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DP IB Maths: AA SL

Revision Notes

Home / IB / Maths: AA SL / DP / Revision Notes / 3. Geometry & Trigonometry / 3.4 Further Trigonometry / 3.4.1 The Unit Circle


3.4.1 The Unit Circle


Defining Sin, Cos and Tan

What is the unit circle?

  • The unit circle is a circle with radius 1 and centre (0, 0)
  • Angles are always measured from the positive x-axis and turn:
    • anticlockwise for positive angles
    • clockwise for negative angles
  • It can be used to calculate trig values as a coordinate point (x, y) on the circle
    • Trig values can be found by making a right triangle with the radius as the hypotenuse
    • Where θ is the angle measured anticlockwise from the positive x-axis
    • The x-axis will always be adjacent to the angle, θ
  • SOHCAHTOA can be used to find the values of sinθ, cosθ and tanθ easily
  • As the radius is 1 unit
    • the x coordinate gives the value of cosθ
    • the y coordinate gives the value of sinθ
  • As the origin is one of the end points - dividing the y coordinate by the x coordinate gives the gradient
    • the gradient of the line gives the value of tanθ
  • It allows us to calculate sin, cos and tan for angles greater than 90° (begin mathsize 16px style straight pi over 2 end stylerad)

ib-aa-sl-the-unit-circle-diagram-1

Worked Example

The coordinates of a point on a unit circle, to 3 significant figures, are (0.629, 0.777). Find θ° to the nearest degree.

efewCfDn_aa-sl-3-4-1-defining-sin-and-cos-we-solution-1

Using The Unit Circle

What are the properties of the unit circle?

  • The unit circle can be split into four quadrants at every 90° (begin mathsize 16px style straight pi over 2 end style rad)
    • The first quadrant is for angles between 0 and 90° 
      • All three of Sinθ, Cosθ and Tanθ are positive in this quadrant
    • The second quadrant is for angles between 90° and 180° (begin mathsize 16px style straight pi over 2 end style rad and begin mathsize 16px style straight pi end style rad)
      • Sinθ is positive in this quadrant
    • The third quadrant is for angles between 180° and 270° (begin mathsize 16px style straight pi end style rad and begin mathsize 16px style fraction numerator 3 straight pi over denominator 2 end fraction end style)
      • Tanθ is positive in this quadrant
    • The fourth quadrant is for angles between 270° and 360° (begin mathsize 16px style fraction numerator 3 straight pi over denominator 2 end fraction end style rad and begin mathsize 16px style 2 straight pi end style)
      • Cosθ is positive in this quadrant
    • Starting from the fourth quadrant (on the bottom right) and working anti-clockwise the positive trig functions spell out CAST
      • This is why it is often thought of as the CAST diagram
      • You may have your own way of remembering this
      • A popular one starting from the first quadrant is All Students Take Calculus
    • To help picture this better try sketching all three trig graphs on one set of axes and look at which graphs are positive in each 90° section

How is the unit circle used to find secondary solutions?

  • Trigonometric functions have more than one input to each output
    • For example sin 30° = sin 150° = 0.5
    • This means that trigonometric equations have more than one solution
    • For example both 30° and 150° satisfy the equation sin x = 0.5
  • The unit circle can be used to find all solutions to trigonometric equations in a given interval
    • Your calculator will only give you the first solution to a problem such as x = sin-1(0.5)
      • This solution is called the primary value
    • However, due to the periodic nature of the trig functions there could be an infinite number of solutions
      • Further solutions are called the secondary values
    • This is why you will be given a domain in which your solutions should be found
      • This could either be in degrees or in radians
      • If you see π or some multiple of π then you must work in radians
  • The following steps may help you use the unit circle to find secondary values

STEP 1: Draw the angle into the first quadrant using the x or y coordinate to help you

  • If you are working with sin x = k, draw the line from the origin to the circumference of the circle at the point where the y coordinate is k
  • If you are working with cos x = k, draw the line from the origin to the circumference of the circle at the point where the x coordinate is k
  • If you are working with tan x = k, draw the line from the origin to the circumference of the circle such that the gradient of the line is k
    • This will give you the angle which should be measured from the positive x-axis…
      • … anticlockwise for a positive angle
      • … clockwise for a negative angle

STEP 2: Draw the radius in the other quadrant which has the same...

  • ... x-coordinate if solving cos x = k
    • This will be the quadrant which is vertical to the original quadrant
  • ... y-coordinate if solving sin x = k
    • This will be the quadrant which is horizontal to the original quadrant
  • ... gradient if solving tan x = k
    • This will be the quadrant diagonal to the original quadrant

STEP 3: Work out the size of the second angle, measuring from the positive x-axis

  • … anticlockwise for a positive angle
  • … clockwise for a negative angle
    • You should look at the given range of values to decide whether you need the negative or positive angle

STEP 4: Add or subtract either 360° or 2π radians to both values until you have all solutions in the required range

aa-sl-3-4-1-using-the-unit-circle-diagram-1

Exam Tip

  • Being able to sketch out the unit circle and remembering CAST can help you to find all solutions to a problem in an exam question 

Worked Example

Given that one solution of cosθ = 0.8 is θ = 0.6435 radians correct to 4 decimal places, find all other solutions in the range -2π ≤ θ ≤ 2π.  Give your answers correct to 3 significant figures.

aa-sl-3-4-1-using-the-unit-circle-we-solution-2



  • 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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