Basic Identities Involving Sines, Cosines, and Tangents

1. Section 4-4
Basic Identities Involving Sines, Cosines, and
Tangents

2. Identity
An equation that is true for all possible values of the
variable

3. Example 1
Complete the following in your calculator.
⎛ 3π ⎞ 2 ⎛ 3π ⎞
sin ⎜ ⎟ + cos ⎜ ⎟
2
cos 30° + sin 30°
2 2
⎝4⎠ ⎝4⎠
1 1
sin 2 ( −25° ) + cos 2 ( −25° ) cos 2 ( 4π ) + sin 2 ( 4π )
1 1

4. Pythagorean Identity
For all theta,
cos (θ ) + sin (θ ) = 1
2 2

5. Example 2
If sinθ = , find cosθ.
1
3
sin θ + cos θ = 1
2 2
()
2
+ cos θ = 1
2
1
3
−
− 1
1
9
9
cos θ =
2 8
9
cos θ = ±
2 8
9
cosθ = ± 8
3
cosθ = ± 22
3

6. Opposites Theorem
For all theta,
()
cos −θ = cosθ
sin ( −θ ) = − sinθ
tan ( −θ ) = − tanθ

7. Example 3
⎛ π⎞ ⎛π⎞
2
3
( ) b.sin ⎜ − ⎟ = − . Find − sin ⎜ ⎟
a.cos30° = . Find cos −30°
⎝ 4⎠ ⎝ 4⎠
2
2
2
3
−
2 2

8. Supplements Theorem
For all theta in radians,
()
sin π − θ = sinθ
cos (π − θ ) = − cosθ
tan (π − θ ) = − tanθ

9. Complements Theorem
For all theta in radians,
⎛π ⎞
sin ⎜ − θ ⎟ = cosθ
⎝2 ⎠
⎛π ⎞
cos ⎜ − θ ⎟ = sinθ
⎝2 ⎠

10. Example 4
() ( )
If sin x = .681, find sin -x and sin π - x .
()
sin -x = −.681
sin (π − x ) = .681

11. Half-turn Theorem
For all theta in radians,
()
cos π + θ = − cosθ
sin (π + θ ) = − sinθ
tan (π + θ ) = tanθ

12. Example 5
Using the unit circle, explain why sin (π − θ ) = sinθ for all θ .
On the unit circle, π = 180° . When you measure theta,
you start at 0°. So, you’re beginning at points that are
reflections of each other. As you plot the values, you will
notice they remain as reflections over the y-axis, which
will keep the y-coordinates the same, which is sinθ .

13. Homework
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