Precalculus Examples

$\sin (360 ° + θ) = \sin (θ)$

Step 1

Start on the left side.

$\sin (360 ° + θ)$

Step 2

Rearrange terms.

$\sin (θ + 360 °)$

Step 3

Double negation

$\sin (θ - (- 360))$

Step 4

Apply the difference of angles identity.

$\sin (θ) \cos (- 360) - \cos (θ) \sin (- 360)$

Step 5

Simplify the expression.

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Step 5.1

Simplify each term.

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Step 5.1.1

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\sin (θ) \cos (0) - \cos (θ) \sin (- 360)$

Step 5.1.2

The exact value of $\cos (0)$ is $1$ .

$\sin (θ) \cdot 1 - \cos (θ) \sin (- 360)$

Step 5.1.3

Multiply $\sin (θ)$ by $1$ .

$\sin (θ) - \cos (θ) \sin (- 360)$

Step 5.1.4

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\sin (θ) - \cos (θ) \sin (0)$

Step 5.1.5

The exact value of $\sin (0)$ is $0$ .

$\sin (θ) - \cos (θ) \cdot 0$

Step 5.1.6

Multiply $- \cos (θ) \cdot 0$ .

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Step 5.1.6.1

Multiply $0$ by $- 1$ .

$\sin (θ) + 0 \cos (θ)$

Step 5.1.6.2

Multiply $0$ by $\cos (θ)$ .

$\sin (θ) + 0$

Step 5.2

Add $\sin (θ)$ and $0$ .

$\sin (θ)$

Step 6

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sin (360 ° + θ) = \sin (θ)$ is an identity