Download Lesson 4: Evaluating Trigonometric Ratios for Any Angle Between

Lesson 4: Evaluating Trigonometric Ratios for Any Angle Between
0 and 360 (Part 1)
Part A – Definitions
 CAST rule: An easy way to indicate the trigonometric ratios that are
POSITIVE in the four quadrants.
Example 1: Determining the trigonometric ratio given an obtuse angle.
Procedure:
 find the related acute angle
 use special triangles, to determine the exact ratio
 go back to the original, principal angle and use the CAST rule to determine
whether your answer is positive or negative
NOTE: The CAST rule helps you to remember which trigonometric ratios are
positive in each quadrant.
S A
T C
Evaluate. Leave answer as an exact value where appropriate.
a) tan 150
b) sin 225
c) cot 120
d) cos 330
e) csc 240
f) sin 315
Part B – Trig. Ratios for Angles Between 0 and 360
Key Idea:
The trig. ratios for any principal angle, θ , in standard position, where 0  θ  360 ,
can be determined by finding the related acute angle using coordinates of any point
P (x , y ) that lies on the terminal arm of the angle.
Example 2:
The point P (3, 4) lies on the terminal arm of an angle θ in standard position.
a) Sketch the angle.
b) Determine the exact ratio for sin θ , cosθ , and tan θ .
c) Repeat the above steps for points (3, 4) , (3,  4) , and (3,  4) .
QII
Sine
y
QI
All
(3, 4)
θ
x
Tangent
QIII
Cosine
QIV
Example 3:
The P (5,  12) lies on the terminal arm of angle θ in standard position.
a) Determine the exact values of sin θ , cosθ , and tanθ .
b) State the value of θ .
y
x
Worksheet: Trig. Ratios for Angles Between 0 and 360
Evaluate. Leave answer as an exact value where appropriate.
a) cos155
b) sin 214
c) tan 315
d) cos 240
e) cot 150
f) cos 225
g) sin 300
h) sec 330
Answers:
a)  0.9063
b)  0.5592
c) -1
e)  3
f) 
2
2
g) 
d) 
3
2
h)
1
2
2
3