The Unit Circle (DP IB Analysis & Approaches (AA): HL): Revision Note

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

How can the unit circle be used to define sin, cos and tan?

• Measure the angle $\theta$ from the positive x-axis

• Label the point on the circle $\left(x,y\right)$

• The trigonometric ratios can be found using SOHCAHTOA

  • $\sin \theta$ is the y-coordinate of the point

  • $\cos \theta$ is the x-coordinate of the point

  • $\tan \theta$ is the gradient of the line segment from the centre to the point

• You can use these definitions to calculate sin, cos and tan for angles that are:

  • greater than 90°

  • negative

    • measure the angle clockwise

  • greater than 360°

    • go round the circle again

How can I use the unit circle to construct the sine graph?

• The y-coordinates of the points on the unit circle give the values of sine

• You can plot the y-coordinates against the angles to form the sine graph

• You should remember the general shape

  • The sine graph starts at 0

  • It then increases and is equal to 1 when the angle is $\frac{\mathrm{\pi }}{2}$

  • It then decreases and returns to 0 when the angle is $\mathrm{\pi }$

  • It continues to decrease and is equal to -1 when the angle is $\frac{3\mathrm{\pi }}{2}$

  • It then increases and returns to 0 when the angle is $2\mathrm{\pi }$

  • The graph then repeats itself

How can I use the unit circle to construct the cosine graph?

• The x-coordinates of the points on the unit circle give the values of cosine

• You can plot the x-coordinates against the angles to form the cosine graph

• You should remember the general shape

  • The cosine graph starts at 1

  • It then decreases and is equal to 0 when the angle is $\frac{\mathrm{\pi }}{2}$

  • It continues to decrease and is equal to -1 when the angle is $\mathrm{\pi }$

  • It then increases and is returns to 0 when the angle is $\frac{3\mathrm{\pi }}{2}$

  • It continues to increase and returns to 1 when the angle is $2\mathrm{\pi }$

  • The graph then repeats itself

Using the unit circle

When are sin, cos and tan positive?

• The unit circle can be split into four quadrants at every 90°

  • 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°

    • sinθ is positive in this quadrant

  • The third quadrant is for angles between 180° and 270°

    • tanθ is positive in this quadrant

  • The fourth quadrant is for angles between 270° and 360°

    • cosθ is positive in this quadrant

What are the symmetries of the unit circle?

• You can find symmetries by comparing the points on the unit circles which form the angles

  • $\theta$

  • $180-\theta$

  • $180+\theta$

  • $360-\theta$

• Adding 360° does not change the trig ratio

  • $\sin \left(360+\theta \right)=\sin \theta$

  • $\cos \left(360+\theta \right)=\cos \theta$

  • $\tan \left(360+\theta \right)=\tan \theta$

• $-\theta$ gives the same ratios as $360-\theta$

  • $\sin \left(-\theta \right)=-\sin \theta$

  • $\cos \left(-\theta \right)=\cos \theta$

  • $\tan \left(-\theta \right)=-\tan \theta$

How can I find the value of sin, cos and tan using the symmetries of the unit circle?

• You can write a trig ratio of any angle in terms of a trig ratio of an acute angle

• Compare the angle to 180° or 360° and use the symmetries

• For example: 210° = 180° + 30°

  • $\sin 210=-\sin 30$

  • $\cos 210=-\cos 30$

  • $\tan 210=\tan 30$

• For example: 315° = 360° - 45°

  • $\sin 315=-\sin 45$

  • $\cos 315=\cos 45$

  • $\tan 315=-\tan 45$

How can I use the unit circle to find solutions to simple trigonometric equations?

• STEP 1
  Use the inverse trig buttons on your calculator to find the principal value

  • e.g. to solve $\sin x=0.5$, type in ${\sin }^{-1}\left(0.5\right)$

• STEP 2
  Label this point on the unit circle and draw the radius

• STEP 3
  Find the second point on the unit circle that also leads to a solution and draw the radius

  • For the equation $\sin x=k$

    • The second point has the same y-coordinate

    • It is in the quadrant horizontal to the quadrant with the first radius

  • For the equation $\cos x=k$

    • The second point has the same x-coordinate

    • It is in the quadrant vertical to the quadrant with the first radius

  • For the equation $\tan x=k$

    • The second point has a radius with the same gradient

    • It is in the quadrant diagonal to the quadrant with the first radius

• STEP 4
  Find the angle from the positive x-axis to the second radius using symmetries

  • e.g. $\sin \left(30\right)=\sin \left(150\right)$ and $\sin \left(-60\right)=\sin \left(-120\right)$

  • e.g. $\cos \left(30\right)=\cos \left(-30\right)$ and $\cos \left(120\right)=\cos \left(240\right)$

  • e.g. $\tan \left(30\right)=\tan \left(210\right)$ and $\tan \left(-60\right)=\tan \left(120\right)$

• STEP 5
  Use the two angles to find all solutions in the given interval

  • Add or subtract multiples of 360° from the two angles