MHF4U
Trig Ratios and Special Triangles
Recall: We use two special triangles to find the exact values of sine, cosine and tangent for angles
30°, 45° and 60°. Now we have to convert this to radians!
(Side note: How do we switch calculators to radians?)
Remember – 1.5 rad does NOT equal 1.5°
Special Angle Triangles – in Radians!
In addition to special triangles, there are some key radian measures to remember:
Now instead of solving for trigonometric expressions and equations in degrees, we can solve
using radians.
Steps:
1. In general, to find trig ratios for angles greater than 90° (____); you need to find the related
acute angles (The related acute angle is the angle formed by the terminal arm of an angle in
standard position and the 𝑥-axis.)
2. The CAST rule can be used to determine the signs of the trig functions
4𝜋
Find the exact values of sin
Example 2
Calculate the 6 trig ratios for
Example 3
Evaluate. (Assume that we want exact answers unless otherwise stated).
𝜋
𝜋
𝜋
𝜋
𝜋
a) cot 4
b) csc 6
c) cos 4 sin 4 − 2 tan2 6
3
and cos
4𝜋
Example 1
3
11𝜋
6
When working with trig ratios when the terminal arm is on an axis, think about the function
𝑓(𝑥) = sin 𝑥.
𝜋
Example 4
Find the exact value of sin 2 .
Example 5
Evaluate.
a) cot
5𝜋
4
+ tan
11𝜋
6
tan
5𝜋
3
𝜋
6
𝜋
6
𝜋
cos2 4
𝜋
2
cos tan +3 sin
b)
Example 6
Find the angle in degrees to 1 decimal place.
a) sin 𝜃 = 0.2273
b) tan 𝐴 = 1.5
c) csc 𝜃 = 2.00
Example 7
Find the angle in radians to 3 decimal places.
a) sin 𝜃 = 0.2273
b) tan 𝐴 = 1.5
c) csc 𝜃 = 2.00
Example 8
If (−2, −3) lies on the terminal arm of an angle θ in standard position:
Determine the exact values of sin 𝜃, cos 𝜃 and tan 𝜃
Calculate the related acute angle in radians to 2 decimal places.
Example 9
𝜋
Kerry is flying a kite with 50 𝑚 of string. The string makes an angle of 6 with the
ground. The wind picks and the kite flies higher so that the string now makes an
𝜋
angle of 3 with the ground. Determine the horizontal distance that the kite moves
between its 2 positions, to one decimal place.