Download 150 Lecture Notes - Section 6.3 Trigonometric Functions of Angles

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Marcia
Drost, March 7, 2008
150 Lecture Notes - Section 6.3
Trigonometric Functions of Angles
Definition of the Trigonometric Functions
Let θ be
p an angle in standard position and let P(x,y) be a point on the terminal side.
If r = x2 + y 2 is the distance from the origin to the point P(x,y), then
y
r
cos θ =
x
r
tan θ =
y
(x 6= 0)
x
r
csc θ = (y 6= 0)
y
sec θ =
r
(x 6= 0)
x
cot θ =
x
(y 6= 0)
y
sin θ =
Reference Angle - Let θ be an angle in standard position. The reference angle θ̄ associated
with θ is the acute angle formed by the terminal side of θ and the x-axis.
Example 1 - Find the reference angle for each of the following:
5π
(a) θ =
3
(b) θ = 870o
Evaluating Trigonometric Functions Using Reference Angles
To find the values of the trigonometric functions for any angle θ, follow these steps:
1. Find the reference angle θ̄ associated with the angle θ.
2. Determine the sign of the trigonometric function of θ.
3. The value of the trigonometric function of θ is the same, except possibly for sign,
as the value of the trigonometric function of θ̄.
Example 2 - Find the value of each of the following.
(a) sin 240o
(b) cot
16π
3
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Drost, March 7, 2008
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Evaluating Trigonometric Functions Using Special Triangles
An alternative method to finding the values of the trigonometric functions for any
angle θ is as follows:
1. Find the reference angle θ̄ associated with the angle θ. (same as above)
2. Draw the appropriate special triangle using the reference angle: use the terminal
side of θ as the hypotenuse and use the angle θ̄ as the acute angle of the right
triangle with vertex at the origin.
3. Fill in the standard side lengths of the special triangle. Adjust the signs of the
adjacent and opposite sides as appropriate to the quadrant the terminal side is
in. The hypotenuse is ALWAYS positive.
Example 3 - Find the value of each of the following.
(a) cos 495o
π
(b) sec −
4
(c) tan 3π
(d) sec
7π
2
3
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Drost, March 7, 2008
Pythagorean Identities
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
1 + cot2 θ = cscθ
Let θ be an acute angle such that sin θ = 0.6
Find cos θ and tan θ.
sin2 θ + cos2 θ = 1
(.6)2 + cos2 θ = 1
cos2 θ = .64
cos θ = .8
.6
.8
3
tan θ = = .75
4
tan θ =
Let θ be an acute angle such that tan θ = 3
θ
Find a) cot θ and b) sec θ
Evaluate: sec (5o 40′ 12”)
40
12
5+
+
= 5.67o
60 3600
sec (5.67o) =
1
≈ 1.00492
cos 5.67o
Reciprocal Identities
1
csc θ =
sin θ
tan θ =
sin θ
cos θ
sec θ =
1
cos θ
cot θ =
cos θ
sin θ
cot θ =
1
tan θ
Example 4 - Use the Pythagorean and reciprocal identities to express tan θ in terms of
sin θ, where θ is in quadrant II.
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Drost, March 7, 2008
Example 5 - Given tan θ =
2
3
and θ in quadrant III, find cos θ.
Area of a Triangle
The area A of a triangle with sides of lengths a and b and with included angle θ is
1
A = ab sin θ
2
Example 6 - Find the area of a triangle with sides of length 7 and 9 and included angle
120o.
Example 7 - Find the area of the shaded region in the figure below if the radius is 8
π
inches, and the central angle is .
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