Precalculus Examples

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

Step 1

Start on the left side.

$\cos (x - \frac{3 π}{2})$

Step 2

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

$0 + \sin (x) \sin (\frac{3 π}{2})$

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.

$0 + \sin (x) (- \sin (\frac{π}{2}))$

Step 3.1.5

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

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

Step 3.1.6

Multiply $- 1$ by $1$ .

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

Step 3.1.7

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

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

Step 3.1.8

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.

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