4.2 Co terminal and Related Angles

In this section, you will learn how to identify different angles that have the same trigonometric ratio, as well as learn how they are related.

To do this you will have to visualize the terminal arm rotating around a circle centred at the origin of a grid with a radius of r. This is done so that we can extend our understanding of trig functions for a broader class of angles and see how different angles are related.

http://www.mathsisfun.com/geometry/unit-circle.html

𝜃x

yr

The circle being used has radius r. The radius and the coordinates of a point on the circle (x, y) are related to the primary trig ratios. Study the circle and write expressions for sin 𝜃, cos 𝜃, and tan 𝜃 in terms of x, y, and r.

(x, y)

DO IT NOW!

sin 𝜃 =

cos 𝜃 =

tan 𝜃 =

When finding the trig ratios of positive angles we are rotating counter clockwise from 0 degrees toward 360. The CAST rule helps us determine which trig ratios are positive in each quadrant

C

AS

T

all

cosine

sine

tan

CAST Rule

Quadrant 1Quadrant 2

Quadrant 3 Quadrant 4

Ex. Given that and that lies in the first quadrant, determine exact values for and

Example 1

Example 2

a) Find both angles of 𝜃 for question 1 between 0 and 360 that have the same sine value

b) Find another angle that will have the same cosine ratio as #1

Example 3: Find both angles between 0° and 360° where sin𝜃  = -0.7

Example 4: Find both angles between 0° and 360° where tanθ=2.1

Example 5: The point (-7, 19) lies on the terminal arm. Find the angle to the terminal arm. Find the reference angle.

Co-terminal Angles: Angles in standard position that have the same terminal arm.

Starting at 30° and rotating 360° counterclockwise will bring you back to the same terminal arm.

30° + 360° = 390°

Therefore 30° and 390° are co-terminal angles.

Note: you can also move clockwise and get negative angles as co-terminal angles.

Example 6: Find three other co-terminal angles of 60°