Trigonometry Examples

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

Step 1

Start on the left side.

$\cos (\frac{3 π}{2} - x)$

Step 2

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

$0 + \sin (\frac{3 π}{2}) \sin (x)$

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.

$0 - \sin (\frac{π}{2}) \sin (x)$

Step 3.1.5

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

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

Step 3.1.6

Multiply $- 1$ by $1$ .

$0 - 1 \sin (x)$

Step 3.1.7

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.

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