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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 3. Geometry & Trigonometry / 3.3 Trigonometry Toolkit / 3.3.1 Pythagoras & Right-Angled Triganometry


3.3.1 Pythagoras & Right-Angled Triganometry


Pythagoras

What is the Pythagorean theorem?

  • Pythagoras’ theorem is a formula that works for right-angled triangles only
  • It states that for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides
    • The hypotenuse is the longest side in a right-angled triangle
      • It will always be opposite the right angle
    • If we label the hypotenuse c, and label the other two sides a and b, then Pythagoras’ theorem tells us that

begin mathsize 22px style a squared space plus space b squared space equals space c squared end style

  • The formula for Pythagoras’ theorem is assumed prior knowledge and is not in the formula booklet
    • You will need to remember it

How can we use Pythagoras’ theorem?

  • If you know two sides of any right-angled triangle you can use Pythagoras’ theorem to find the length of the third side
    • Substitute the values you have into the formula and either solve or rearrange
  • To find the length of the hypotenuse you can use:

begin mathsize 22px style c equals blank square root of a squared plus b squared end root end style

  • To find the length of one of the other sides you can use:

a space equals blank square root of c squared space minus space b squared end root   or  b space equals blank square root of c squared space minus space a squared end root

  • Note that when finding the hypotenuse you should add inside the square root and when finding one of the other sides you should subtract inside the square root
  • Always check your answer carefully to make sure that the hypotenuse is the longest side
  • Note that Pythagoras’ theorem questions will rarely be standalone questions and will often be ‘hidden’ in other geometry questions

What is the converse of the Pythagorean theorem?

  • The converse of the Pythagorean theorem states that if a squared space plus space b squared space equals space c squared  is true then the triangle must be a right-angled triangle
    • This is a very useful way of determining whether a triangle is right-angled
  • If a diagram in a question does not clearly show that something is right-angled, you may need to use Pythagoras’ theorem to check

Exam Tip

  • Pythagoras' theorem pops up in lots of exam questions so bear it in mind whenever you see a right-angled triangle in an exam question!

Worked Example

ABCDEF is a chocolate bar in the shape of a triangular prism.  The end of the chocolate bar is an isosceles triangle where AC = 3 cm and AB = BC = 5 cm.  M is the midpoint of AC. This information is shown in the diagram below.

diagram-for-we-3-3-1-pythag

Calculate the length BM.

3-3-1-ai-sl-pythag-we-solution

Right-Angled Trigonometry

What is Trigonometry?

  • Trigonometry is the mathematics of angles in triangles
  • It looks at the relationship between side lengths and angles of triangles
  • It comes from the Greek words trigonon meaning ‘triangle’ and metron meaning ‘measure’

 

What are Sin, Cos and Tan?

  • The three trigonometric functions Sine, Cosine and Tangent come from ratios of side lengths in right-angled triangles
  • To see how the ratios work you must first label the sides of a right-angled triangle in relation to a chosen angle
    • The hypotenuse, H, is the longest side in a right-angled triangle
      • It will always be opposite the right angle
    • If we label one of the other angles θ, the side opposite θ will be labelled opposite, O, and the side next to θ will be labelled adjacent, A
  • The functions Sine, Cosine and Tangent are the ratios of the lengths of these sides as follows

Sin space theta blank equals space opposite over hypotenuse space equals space O over H

 Cos space theta blank equals space adjacent over hypotenuse space equals space straight A over straight H

Tan space theta blank equals space opposite over adjacent space equals space straight O over straight A

    • These are not in the formula book, you must remember them
  • The mnemonic SOHCAHTOA is often used as a way of remembering which ratio is which
    • Sin is Opposite over Hypotenuse
    • Cos is Adjacent over Hypotenuse
    • Tan is Opposite over Adjacent

Right-Angled Triangles Diagram 1

How can we use SOHCAHTOA to find missing lengths?

  • If you know the length of one of the sides of any right-angled triangle and one of the angles you can use SOHCAHTOA to find the length of the other sides
    • Always start by labelling the sides of the triangle with H, O and A
    • Choose the correct ratio by looking only at the values that you have and that you want
      • For example if you know the angle and the side opposite it (O) and you want to find the hypotenuse (H) you should use the sine ratio
    • Substitute the values into the ratio
    • Use your calculator to find the solution

 

How can we use SOHCAHTOA to find missing angles?

  • If you know two sides of any right-angled triangle you can use SOHCAHTOA to find the size of one of the angles
  • Missing angles are found using the inverse functions:

 theta space equals space Sin to the power of negative 1 end exponent space O over H   ,    theta space equals space Cos to the power of negative 1 end exponent space straight A over straight H   ,   theta space equals space Tan to the power of negative 1 end exponent space straight O over straight A

  • After choosing the correct ratio and substituting the values use the inverse trigonometric functions on your calculator to find the correct answer

Exam Tip

  • You need to remember the sides involved in the different trig ratios as they are not given to you in the exam

Worked Example

Find the values of x and y in the following diagram. Give your answers to 3 significant figures.

sa-diagram-for-we-3-3-1-trig

3-3-1-ai-sl-r-a-trig-we-solution

3D Problems

How does Pythagoras work in 3D?

  • 3D shapes can often be broken down into several 2D shapes
  • With Pythagoras’ Theorem you will be specifically looking for right-angled triangles
    • The right-angled triangles you need will have two known sides and one unknown side
    • Look for perpendicular lines to help you spot right-angled triangles
  • There is a 3D version of the Pythagorean theorem formula:

begin mathsize 22px style d squared space equals space x squared space plus space y squared space plus space z squared end style 

    • However it is usually easier to see a problem by breaking it down into two or more 2D problems

How does SOHCAHTOA work in 3D?

  • Again look for a combination of right-angled triangles that would lead to the missing angle or side
  • The angle you are working with can be awkward in 3D
    • The angle between a line and a plane is not always obvious
    • If unsure put a point on the line and draw a new line to the plane
      • This should create a right-angled triangle

 

3DPythagTrig Notes fig6

Exam Tip

  • Annotate diagrams that are given to you with values that you have calculated
  • It can be useful to make additional sketches of parts of any diagrams that are given to you, especially if there are multiple lengths/angles that you are asked to find
  • If you are not given a diagram, sketch a nice, big, clear one!

Worked Example

A pencil is being put into a cuboid shaped box which has dimensions 3 cm by 4 cm by 6 cm. Find, giving your answers to 1 decimal place:

 

a)
the length of the longest pencil that could fit inside the box,

ai-sl-3-3-1-3d-pythag-trig-we-solution-a

 

b)
the angle that the pencil would make with the top of the box.

ai-sl-3-3-1-3d-pythag-trig-we-solution-b



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 1.2 Exponentials & Logs
            • 1.2.1 Introduction to Logarithms
              • 1.2.2 Laws of Logarithms
                • 1.2.3 Solving Exponential Equations
                • 1.3 Sequences & Series
                  • 1.3.1 Language of Sequences & Series
                    • 1.3.2 Arithmetic Sequences & Series
                      • 1.3.3 Geometric Sequences & Series
                        • 1.3.4 Applications of Sequences & Series
                          • 1.3.5 Compound Interest & Depreciation
                          • 1.4 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 2. Functions
                                                            • 2.1 Linear Functions & Graphs
                                                              • 2.1.1 Equations of a Straight Line
                                                              • 2.2 Quadratic Functions & Graphs
                                                                • 2.2.1 Quadratic Functions
                                                                  • 2.2.2 Factorising & Completing the Square
                                                                    • 2.2.3 Solving Quadratics
                                                                      • 2.2.4 Quadratic Inequalities
                                                                        • 2.2.5 Discriminants
                                                                        • 2.3 Functions Toolkit
                                                                          • 2.3.1 Language of Functions
                                                                            • 2.3.2 Composite & Inverse Functions
                                                                              • 2.3.3 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 3. Geometry & Trigonometry
                                                                                                                    • 3.1 Geometry Toolkit
                                                                                                                      • 3.1.1 Coordinate Geometry
                                                                                                                        • 3.1.2 Radian Measure
                                                                                                                          • 3.1.3 Arcs & Sectors
                                                                                                                          • 3.2 Geometry of 3D Shapes
                                                                                                                            • 3.2.1 3D Coordinate Geometry
                                                                                                                              • 3.2.2 Volume & Surface Area
                                                                                                                              • 3.3 Trigonometry Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 Trigonometry
                                                                                                                                      • 3.4.1 The Unit Circle
                                                                                                                                        • 3.4.2 Exact Values
                                                                                                                                        • 3.5 Trigonometric Functions & Graphs
                                                                                                                                          • 3.5.1 Graphs of Trigonometric Functions
                                                                                                                                            • 3.5.2 Transformations of Trigonometric Functions
                                                                                                                                              • 3.5.3 Modelling with Trigonometric Functions
                                                                                                                                              • 3.6 Trigonometric Equations & Identities
                                                                                                                                                • 3.6.1 Simple Identities
                                                                                                                                                  • 3.6.2 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 4. Statistics & Probability
                                                                                                                                                                                              • 4.1 Statistics Toolkit
                                                                                                                                                                                                • 4.1.1 Sampling & Data Collection
                                                                                                                                                                                                  • 4.1.2 Statistical Measures
                                                                                                                                                                                                    • 4.1.3 Frequency Tables
                                                                                                                                                                                                      • 4.1.4 Linear Transformations of Data
                                                                                                                                                                                                        • 4.1.5 Outliers
                                                                                                                                                                                                          • 4.1.6 Univariate Data
                                                                                                                                                                                                            • 4.1.7 Interpreting Data
                                                                                                                                                                                                            • 4.2 Correlation & Regression
                                                                                                                                                                                                              • 4.2.1 Bivariate Data
                                                                                                                                                                                                                • 4.2.2 Correlation & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 4.5 Binomial Distribution
                                                                                                                                                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                                                                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                                                                                                                • 4.6 Normal Distribution
                                                                                                                                                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                                                                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                                                                                                                                                      • 4.6.3 Standardisation of Normal Variables
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                  • 5.2.3 Higher Order Derivatives
                                                                                                                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                                                                                                                        • 5.3 Integration
                                                                                                                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        Amber Perry

                                                                                                                                                                                                                                                                                                        Author: Amber

                                                                                                                                                                                                                                                                                                        Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.


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