Expand Using Sum/Difference Formulas sin((3pi)/2+x)

$\sin (\frac{3 π}{2} + x)$

Use the sum formula for sine to simplify the expression. The formula states that $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

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

Remove parentheses.

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

Simplify each term.

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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.

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

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

$- 1 \cdot 1 \cos (x) + \cos (\frac{3 π}{2}) \sin (x)$

Multiply $- 1$ by $1$ .

$- 1 \cos (x) + \cos (\frac{3 π}{2}) \sin (x)$

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

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

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

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

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

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

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

$- \cos (x) + 0$

Add $- \cos (x)$ and $0$ .

$- \cos (x)$

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