The Unit Circle (Cambridge International IGCSE Further Maths)

The unit circle

What is the unit circle?

• The unit circle is a circle with radius 1 and centre (0, 0)

• The unit circle can be used to explain how trig functions work with angles that are not acute

• 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° ($\frac{π}{2}$rad)

Worked Example The coordinates of a point on a unit circle, to 3 significant figures, are (0.629, 0.777).

Find θ in radians correct to 2 significant figures..

We know that $( x , y ) = ( \cos \\theta , \sin \\theta )$ so we can use either ratio to solve for $θ$.

$\begin{gathered} \cos \\theta & = 0.629 \\ \\theta & = \cos ^{- 1 } 0.629 \\ = 0.89053... \end{gathered}$

or

$\begin{gathered} \sin \\theta & = 0.777 \\ \\theta & = \sin ^{- 1 } 0.777 \\ = 0.88988... \end{gathered}$

$θ=0.89\mathrm{rad}(2s.f.)$

Using The Unit Circle

What are the properties of the unit circle?

• The unit circle can be split into four quadrants at every 90° ($\frac{π}{2}$ 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° ($\frac{π}{2}$ rad and $π$ rad)

    • Sinθ is positive in this quadrant

  • The third quadrant is for angles between 180° and 270° ($π$ rad and $\frac{3π}{2}$)

    • Tanθ is positive in this quadrant

  • The fourth quadrant is for angles between 270° and 360° ($\frac{3π}{2}$ rad and $2π$)

    • 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

Exam Hint

• 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.

Cosine is positive in the first and fourth quadrants so draw the angle from the horizontal axis in both quadrants.

The primary value is 0.6435 and a secondary value is -0.6435.

*Consider all four possible angles that can be made from the horizontal axis to each of the lines drawn. The range is -2π ≤ θ ≤ 2π so the angles can be in both the positive and negative direction. *

The angle in the positive direction to the line in the fourth quadrant is 2π - 0.6435 = 5.63968... The angle in the negative direction to the line in the first quadrant is -2π + 0.6435 = -5.63968...

Final answer: Solutions are -5.64, -0.644, 0.644, 5.64 rads (3 s.f.)