Trigonometry Examples

$\sin (π + x) + \sin (π - x)$

Step 1

Simplify each term.

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Step 1.1

Apply the sum of angles identity.

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

Step 1.2

Simplify each term.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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.

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

Step 1.2.5

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

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

Step 1.2.6

Multiply $- 1$ by $1$ .

$0 - 1 \sin (x) + \sin (π - x)$

Step 1.2.7

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

$0 - \sin (x) + \sin (π - x)$

Step 1.3

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

$- \sin (x) + \sin (π - x)$

Step 1.4

Apply the difference of angles identity.

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

Step 1.5

Simplify each term.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

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

$- \sin (x) + 0 - \cos (π) \sin (x)$

Step 1.5.4

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.

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

Step 1.5.5

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

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

Step 1.5.6

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

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Step 1.5.6.1

Multiply $- 1$ by $1$ .

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

Step 1.5.6.2

Multiply $- 1$ by $- 1$ .

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

Step 1.5.7

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

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

Step 1.6

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

$- \sin (x) + \sin (x)$

Step 2

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

$0$