Precalculus Examples

$\sin (x + \frac{3 π}{2}) = - \cos (x)$

Step 1

Start on the left side.

$\sin (x + \frac{3 π}{2})$

Step 2

Apply the sum of angles identity.

$\sin (x) \cos (\frac{3 π}{2}) + \cos (x) \sin (\frac{3 π}{2})$

Step 3

Simplify the expression.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$\sin (x) \cos (\frac{π}{2}) + \cos (x) \sin (\frac{3 π}{2})$

Step 3.1.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\sin (x) \cdot 0 + \cos (x) \sin (\frac{3 π}{2})$

Step 3.1.3

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

$0 + \cos (x) \sin (\frac{3 π}{2})$

Step 3.1.4

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

$0 + \cos (x) (- \sin (\frac{π}{2}))$

Step 3.1.5

The exact value of $\sin (\frac{π}{2})$ is $1$ .

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

Step 3.1.6

Multiply $- 1$ by $1$ .

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

Step 3.1.7

Move $- 1$ to the left of $\cos (x)$ .

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

Step 3.1.8

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

$0 - \cos (x)$

Step 3.2

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

$- \cos (x)$

Step 4

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

$\sin (x + \frac{3 π}{2}) = - \cos (x)$ is an identity