Download 12-3 Study Guide and Intervention Trigonometric Functions of

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12-3 Study Guide and Intervention
Trigonometric Functions of General Angles
Trigonometric Functions for General Angles
Trigonometric Functions,
θ in Standard Position
Let θ be an angle in standard position and let P(x, y) be a point on the terminal side of
θ. By the Pythagorean Theorem, the distance r from the origin is given by r = √𝑥 2 + 𝑦 2 .
The trigonometric functions of an angle in standard position may be defined as follows.
sin θ =
csc θ =
𝑦
cos θ =
𝑟
𝑟
𝑦
𝑥
𝑟
,y≠0
𝑦
tan θ =
𝑟
𝑥
,x≠0
𝑥
sec θ = , x ≠ 0
cot θ = , y ≠ 0
𝑥
𝑦
Example: Find the exact values of the six trigonometric functions of θ if the terminal side of θ in standard position
contains the point ( –5, 5√𝟐).
You know that x = –5 and y = 5. You need to find r.
r = √𝑥 2 + 𝑦 2
Pythagorean Theorem
= √(−5)2 + (5√2)2
Replace x with –5 and y with 5√2 .
= √75 or 5√3
Now use x = –5, y = 5√2 , and r = 5√3 to write the six trigonometric ratios.
𝑦
5√2
5√3
=
√6
3
cos θ = 𝑟 = 5
𝑟
5√3
5√2
=
√6
2
sec θ = 𝑥 =
sin θ = 𝑟 =
csc θ = 𝑦 =
𝑥
−5
√3
𝑟
5√3
−5
= −
√3
3
= −√3
𝑦
5√2
−5
= −√2
𝑥
−5
√2
= −
tan θ = 𝑥 =
cot θ = 𝑦 = 5
√2
2
Exercises
The terminal side of θ in standard position contains each point. Find the exact values of the six trigonometric
functions of θ.
1. (8, 4)
2. (4, 4)
3. (0, 4)
4. (6, 2)
Chapter 12
17
Glencoe Algebra 2
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12-3 Study Guide and Intervention (continued)
Trigonometric Functions of General Angles
Trigonometric Functions with Reference Angles If θ is a nonquadrantal angle in standard position, its reference
angle θ ' is defined as the acute angle formed by the terminal side of θ and the x-axis.
Reference
Angle Rule
Example 1: Sketch an angle of measure 205°.
Example 2: Use a reference angle to find the exact
Then find its reference angle.
value of cos
Because the terminal side of 205° lies in Quadrant III,
the reference angle θ ' is 205° – 180° or 25°.
Because the terminal side of
𝟑𝝅
.
𝟒
3𝜋
lies in
4
3𝜋
𝜋
or .
4
4
Quadrant II,
the reference angle θ ' is π –
The cosine function is negative in Quadrant II.
cos
3𝜋
4
𝜋
= – cos 4 = –
√2
2
Exercises
Sketch each angle. Then find its reference angle.
1. 155°
2. 230°
3.
4𝜋
3
4. –
𝜋
6
Find the exact value of each trigonometric function.
5. tan 330°
Chapter 12
6. cos
11𝜋
4
7. cot 30°
18
𝜋
8. csc 4
Glencoe Geometry