Precalculus Examples

$\sin (3 π - x) = \sin (x)$

Step 1

Start on the left side.

$\sin (3 π - x)$

Step 2

Apply the difference of angles identity.

$\sin (3 π) \cos (x) - \cos (3 π) \sin (x)$

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 $2 π$ .

$\sin (π) \cos (x) - \cos (3 π) \sin (x)$

Step 3.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin (0) \cos (x) - \cos (3 π) \sin (x)$

Step 3.1.3

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

$0 \cos (x) - \cos (3 π) \sin (x)$

Step 3.1.4

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

$0 - \cos (3 π) \sin (x)$

Step 3.1.5

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

$0 - \cos (π) \sin (x)$

Step 3.1.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

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

Step 3.1.7

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

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

Step 3.1.8

Multiply $- (- 1 \cdot 1)$ .

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Step 3.1.8.1

Multiply $- 1$ by $1$ .

$0 - - 1 \sin (x)$

Step 3.1.8.2

Multiply $- 1$ by $- 1$ .

$0 + 1 \sin (x)$

Step 3.1.9

Multiply $\sin (x)$ by $1$ .

$0 + \sin (x)$

Step 3.2

Add $0$ and $\sin (x)$ .

$\sin (x)$

Step 4

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

$\sin (3 π - x) = \sin (x)$ is an identity