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DP IB Maths: AI SL

Revision Notes

Home / IB / Maths: AI SL / DP / Revision Notes / 3. Geometry & Trigonometry / 3.3 Trigonometry / 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 Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Exponents & Logarithms
          • 1.1.3 Approximation & Estimation
            • 1.1.4 GDC: Solving Equations
            • 1.2 Sequences & Series
              • 1.2.1 Language of Sequences & Series
                • 1.2.2 Arithmetic Sequences & Series
                  • 1.2.3 Geometric Sequences & Series
                    • 1.2.4 Applications of Sequences & Series
                    • 1.3 Financial Applications
                      • 1.3.1 Compound Interest & Depreciation
                        • 1.3.2 Amortisation & Annuities
                      • 2. Functions
                        • 2.1 Linear Functions & Graphs
                          • 2.1.1 Equations of a Straight Line
                          • 2.2 Further Functions & Graphs
                            • 2.2.1 Functions
                              • 2.2.2 Graphing Functions
                                • 2.2.3 Properties of Graphs
                                • 2.3 Modelling with Functions
                                  • 2.3.1 Linear & Piecewise Models
                                    • 2.3.2 Quadratic & Cubic Models
                                      • 2.3.3 Exponential Models
                                        • 2.3.4 Direct & Inverse Variation
                                          • 2.3.5 Sinusoidal Models
                                            • 2.3.6 Strategy for Modelling Functions
                                          • 3. Geometry & Trigonometry
                                            • 3.1 Geometry Toolkit
                                              • 3.1.1 Coordinate Geometry
                                                • 3.1.2 Arcs & Sectors
                                                • 3.2 Geometry of 3D Shapes
                                                  • 3.2.1 3D Coordinate Geometry
                                                    • 3.2.2 Volume & Surface Area
                                                    • 3.3 Trigonometry
                                                      • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                        • 3.3.2 Non Right-Angled Trigonometry
                                                          • 3.3.3 Applications of Trigonometry & Pythagoras
                                                          • 3.4 Voronoi Diagrams
                                                            • 3.4.1 Voronoi Diagrams
                                                              • 3.4.2 Toxic Waste Dump Problem
                                                            • 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 Coefficients
                                                                                  • 4.2.3 Linear Regression
                                                                                  • 4.3 Probability
                                                                                    • 4.3.1 Probability & Types of Events
                                                                                      • 4.3.2 Conditional Probability
                                                                                        • 4.3.3 Sample Space Diagrams
                                                                                        • 4.4 Probability Distributions
                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                            • 4.4.2 Expected Values
                                                                                            • 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
                                                                                                    • 4.7 Hypothesis Testing
                                                                                                      • 4.7.1 Hypothesis Testing
                                                                                                        • 4.7.2 Chi-squared Test for Independence
                                                                                                          • 4.7.3 Goodness of Fit Test
                                                                                                            • 4.7.4 The t-test
                                                                                                          • 5. Calculus
                                                                                                            • 5.1 Differentiation
                                                                                                              • 5.1.1 Introduction to Differentiation
                                                                                                                • 5.1.2 Applications of Differentiation
                                                                                                                  • 5.1.3 Modelling with Differentiation
                                                                                                                  • 5.2 Integration
                                                                                                                    • 5.2.1 Trapezoid Rule: Numerical Integration
                                                                                                                      • 5.2.2 Introduction to Integration
                                                                                                                        • 5.2.3 Applications of Integration
                                                                                                                      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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