The Unit Circle (DP IB Applications & Interpretation (AI)): Revision Note

Did this video help you?

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

    • 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

How is the unit circle used to construct the graphs of sine and cosine?

  • On the unit circle the y-coordinates give the value of sine

    • Plot the y-coordinate from the unit circle as the y-coordinate on a trig graph for x-coordinates of θ = 0, π/2, π, 3π/2 and 2π

    • Join these points up using a smooth curve

      • To get a clearer idea of the shape of the curve the points for x-coordinates of θ = π/4, 3π/4, 5π/4 and 7π/4 could also be plotted

3-4-1-ib-ai-hl-unit-circle-sine-graph-diagram-1
  • On the unit circle the x-coordinates give the value of cosine

    • Plot the x-coordinate from the unit circle as the y-coordinate on a trig graph for x-coordinates of θ = 0, π/4, π/2, 3π/4 and 2π

    • Join these points up using a smooth curve

      • To get a clearer idea of the shape of the curve the points for x-coordinates of θ = π/4, 3π/4, 5π/4 and 7π/4 could also be plotted

3-4-1-ib-ai-hl-unit-circle-cosine-graph-diagram-2
  • Looking at the unit circle alongside of the sine or cosine graph will help to visualise this clearer

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

Did this video help you?

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

    • Note that whilst this method works for tan, it is complicated and generally unnecessary, tan x repeats every 180° (π radians) so the quickest method is just to add or subtract multiples of 180° to the primary value

  • 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 diagonally across from 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

Examiner Tips and Tricks

  • 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
👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Did this page help you?

Amber

Author: Amber

Expertise: Maths Content Creator

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.

Download notes on The Unit Circle