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DP IB Maths: AA SL

Revision Notes

Home / IB / Maths: AA SL / DP / Revision Notes / 3. Geometry & Trigonometry / 3.6 Trigonometric Equations & Identities / 3.6.1 Simple Identities


3.6.1 Simple Identities


Simple Identities

What is a trigonometric identity?

  • Trigonometric identities are statements that are true for all values of x or theta
  • They are used to help simplify trigonometric equations before solving them
  • Sometimes you may see identities written with the symbol ≡
    • This means 'identical to'

What trigonometric identities do I need to know?

  • The two trigonometric identities you must know are
    • tan space theta space equals space fraction numerator sin space theta over denominator cos space theta end fraction
      • This is the identity for tan θ
    • sin squared theta space plus space cos squared theta space equals space 1
      • This is the Pythagorean identity
      • Note that the notation sin space squared theta is the same as left parenthesis sin space theta right parenthesis to the power of space 2 end exponent
  • Both identities can be found in the formula booklet
  • Rearranging the second identity often makes it easier to work with
    • sin squared theta equals blank 1 minus space cos to the power of 2 space end exponent theta
    • cos squared theta equals blank 1 minus space sin squared theta

Where do the trigonometric identities come from?

  • You do not need to know the proof for these identities but it is a good idea to know where they come from
  • From SOHCAHTOA we know that
    • begin mathsize 16px style sin space theta blank equals opposite over hypotenuse equals O over H end style
    • begin mathsize 16px style cos space theta blank equals adjacent over hypotenuse equals A over H end style
    • begin mathsize 16px style tan space theta blank equals opposite over adjacent equals O over A end style
  • The identity for begin mathsize 16px style tan space theta end style can be seen by diving begin mathsize 16px style sin space theta end style by begin mathsize 16px style cos space theta end style?
    • fraction numerator sin space theta over denominator cos space theta end fraction equals fraction numerator O over H over denominator A over H end fraction equals O over A equals tan space theta
  • The Pythagorean identity can be seen by considering a right-triangle with a hypotenuse of 1
    • Then (opposite)2 + (adjacent)2 = 1
    • Therefore begin mathsize 16px style sin squared space theta plus cos squared space theta blank equals blank 1 end style
  • Considering the equation of the unit circle also shows the Pythagorean identity
    • The equation of the unit circle is begin mathsize 16px style space x squared space plus space y squared space equals space 1 end style
    • The coordinates on the unit circle are begin mathsize 16px style left parenthesis cos space theta comma blank sin space theta right parenthesis end style
    • Therefore the equation of the unit circle could be written begin mathsize 16px style cos squared space theta plus sin squared space theta equals 1 end style
  • A third very useful identity is begin mathsize 16px style sin space theta equals cos space left parenthesis 90 degree minus blank theta right parenthesis blank end styleor begin mathsize 16px style sin space theta equals cos space left parenthesis pi over 2 minus blank theta right parenthesis end style
    • This is not included in the formula booklet but is useful to remember

How are the trigonometric identities used?

  • Most commonly trigonometric identities are used to change an equation into a form that allows it to be solved
  • They can also be used to prove further identities such as the double angle formulae

Exam Tip

  • If you are asked to show that one thing is identical (≡) to another, look at what parts are missing –  for example, if tan x has gone it must have been substituted

Worked Example

Show that the equation 2 sin squared space x minus cos space x equals 0 can be written in the form a cos squared space x plus b cos space x plus c equals 0, where a, b and c are integers to be found.

aa-sl-3-6-1-we-solutions-1



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
        • 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 Proof & Reasoning
                          • 1.4.1 Proof
                          • 1.5 Binomial Theorem
                            • 1.5.1 Binomial Theorem
                          • 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 Graphing Functions
                                              • 2.4 Further Functions & Graphs
                                                • 2.4.1 Reciprocal & Rational Functions
                                                  • 2.4.2 Exponential & Logarithmic Functions
                                                    • 2.4.3 Solving Equations
                                                      • 2.4.4 Modelling with Functions
                                                      • 2.5 Transformations of Graphs
                                                        • 2.5.1 Translations of Graphs
                                                          • 2.5.2 Reflections of Graphs
                                                            • 2.5.3 Stretches of Graphs
                                                              • 2.5.4 Composite Transformations of Graphs
                                                            • 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
                                                                          • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                            • 3.3.2 Non Right-Angled Trigonometry
                                                                              • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                              • 3.4 Further 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 Double Angle Formulae
                                                                                              • 3.6.3 Relationship Between Trigonometric Ratios
                                                                                                • 3.6.4 Linear Trigonometric Equations
                                                                                                  • 3.6.5 Quadratic Trigonometric Equations
                                                                                                • 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 Tranformations 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 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
                                                                                                                                        • 4.6.3 Standardisation of Normal Variables
                                                                                                                                      • 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 Second 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
                                                                                                                                                                        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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