Trigonometry Examples

$\sin (2 π - x) = - \sin (x)$

Step 1

Start on the left side.

$\sin (2 π - x)$

Step 2

Apply the difference of angles identity.

$\sin (2 π) \cos (x) - \cos (2 π) \sin (x)$

Step 3

Simplify the expression.

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Step 3.1

Simplify each term.

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Step 3.1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (0) \cos (x) - \cos (2 π) \sin (x)$

Step 3.1.2

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

$0 \cos (x) - \cos (2 π) \sin (x)$

Step 3.1.3

Multiply $0$ by $\cos (x)$ .

$0 - \cos (2 π) \sin (x)$

Step 3.1.4

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$0 - \cos (0) \sin (x)$

Step 3.1.5

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

$0 - 1 \cdot 1 \sin (x)$

Step 3.1.6

Multiply $- 1$ by $1$ .

$0 - 1 \sin (x)$

Step 3.1.7

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

$0 - \sin (x)$

Step 3.2

Subtract $\sin (x)$ from $0$ .

$- \sin (x)$

Step 4

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

$\sin (2 π - x) = - \sin (x)$ is an identity