Section 4-4Basic Identities Involving Sines, Cosines, and

Tangents

Identity

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

Example 1Complete the following in your calculator.

cos2 30° + sin2 30° sin2 3π4

⎛⎝⎜

⎞⎠⎟+ cos2 3π

4⎛⎝⎜

⎞⎠⎟

sin2 −25°( ) + cos2 −25°( ) cos2 4π( ) + sin2 4π( )

1 1

1 1

Pythagorean Identity

For all theta,

cos2 θ( ) + sin2 θ( ) = 1

Example 2 If sinθ = 1

3, find cosθ.

sin2θ + cos2θ = 1

13( )2

+ cos2θ = 1

− 1

9 − 1

9

cos2θ = 8

9

cos2θ = ± 8

9

cosθ = ± 8

3

cosθ = ± 2 2

3

Opposites TheoremFor all theta,

cos −θ( ) = cosθ

sin −θ( ) = − sinθ

tan −θ( ) = − tanθ

Example 3

a.cos30° =

32

. Find cos −30°( ) b.sin −

π4

⎛⎝⎜

⎞⎠⎟= −

22

. Find − sinπ4

⎛⎝⎜

⎞⎠⎟

32

−2

2

Supplements TheoremFor all theta in radians,

sin π −θ( ) = sinθ

cos π −θ( ) = − cosθ

tan π −θ( ) = − tanθ

Complements TheoremFor all theta in radians,

sinπ2−θ

⎛⎝⎜

⎞⎠⎟= cosθ

cosπ2−θ

⎛⎝⎜

⎞⎠⎟= sinθ

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

sin -x( ) = −.681

sin π − x( ) = .681

Half-turn TheoremFor all theta in radians,

cos π +θ( ) = − cosθ

sin π +θ( ) = − sinθ

tan π +θ( ) = tanθ

Example 5Using the unit circle, explain why for all .

sin π −θ( ) = sinθ θ

On the unit circle, . When you measure theta, you start at . 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 .

π = 180° 0°

sinθ

Homework

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