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Math 4 Pre-Calculus
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Trigonometric Functions Values — 6.2 & 6.4
Reference Angle
Let θ be a nonquadrantal angle in standard position. The reference angle for θ is the acute angle θ R that the terminal side of
θ makes with the x-axis.
The reference angles are illustrated below for positive angles.
Quadrant I
θR = θ
1.
Quadrant II
θR = 1 8 0 ° − θ
Quadrant III
θR = θ − 1 8 0 °
Quadrant IV
θR = 3 6 0 ° − θ
θR = π − θ
θR = θ − π
θ R = 2π − θ
Find the reference angle θ R if θ has the given measure.
a)
θ = 165°
5π
b)
θ =
3
c)
θ = −315°
9π
θ =−
d)
4
Trigonometric Functions for Any Angle
Consider an acute angle θ , in standard position. Let r denote the distance between the origin and a point P ( x, y ) that is on the
terminal side of θ . The following picture illustrates this concept:
Let θ be an angle in standard position on a rectangular coordinate system, and let P ( x, y ) be any point other than the origin
on the terminal side of θ . If d ( O, P ) = r =
x 2 + y 2 (note that r is always positive), then,
sinθ =
y
r
cosθ =
x
r
tanθ =
y
, if x ≠ 0
x
cscθ =
r
, if y ≠ 0
y
s e cθ =
r
, if x ≠ 0
x
cotθ =
x
, if y ≠ 0
y
then
sinθ = y
NOTE: When r = 1 ,
and
cosθ = x
2.
Find the exact values of the six trigonometric functions of θ if θ is in standard position and P is on the
terminal side.
P ( − 8, − 1 5 )
3.
Find the exact values of sine, cosine and tangent of each angle whenever possible. Use r = 1 .
a)
0 ° or 0
4.
π
b)
9 0 ° or
c)
1 8 0 ° or π
d)
2 7 0 ° or
e)
3 6 0 ° or 2π
2
3π
2
Find the exact values of sine, cosine and tangent of each angle whenever possible. Use r = 1 .
π
a)
4 5 ° or
b)
1 3 5 ° or
3π
4
c)
2 2 5 ° or
5π
4
d)
3 1 5 ° or
7π
4
4
5.
6.
Find the exact values of sine, cosine and tangent of each angle whenever possible. Use r = 1 .
π
a)
3 0 ° or
b)
1 5 0 ° or
5π
6
c)
2 1 0 ° or
7π
6
d)
3 3 0 ° or
1 1π
6
6
Find the exact values of sine, cosine and tangent of each angle whenever possible. Use r = 1 .
a)
6 0 ° or
π
3
b)
1 2 0 ° or
2π
3
c)
2 4 0 ° or
4π
3
d)
3 0 0 ° or
5π
3
Theorem on Reference Angles
If θ is a nonquadrantal angle in standard position, then to find the value of a trigonometric function at θ , find its value for the
reference angle θ R and prefix the appropriate sign.
Positive Trigonometric Functions
Quadrant II
Sine
Cosecant
Smart
Quadrant I
All
A
Quadrant III
Quadrant IV
Tangent
Cosine
Cotangent
Secant
Trig
Class
7.
Find the exact value.
a)
s i n ( − 1 2 0 °)
b)
 2π 
sec

 3 
c)
 3π 
cot  −

 4 
d)
c s c ( − 3 0 °)
8.
Use the fundamental identities to find the values of the trigonometric functions for the given conditions.
3
cotθ =
and
cosθ < 0
4
9.
Use the fundamental identities to find the values of the trigonometric functions for the given conditions.
1
and
cosθ = −
sinθ < 0
3
10.
Approximate to three decimal places.
a)
c o s ( 2 . 5)
b)
c s c (1 8 5 ° 1 2′ 4 4′′ )