Precalculus Examples

$\cos (\frac{3 π}{2} + θ) = \sin (θ)$

Step 1

Start on the left side.

$\cos (\frac{3 π}{2} + θ)$

Step 2

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

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

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.

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

Step 3.1.2

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

$0 \cos (θ) - \sin (\frac{3 π}{2}) \sin (θ)$

Step 3.1.3

Multiply $0$ by $\cos (θ)$ .

$0 - \sin (\frac{3 π}{2}) \sin (θ)$

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.

$0 - - \sin (\frac{π}{2}) \sin (θ)$

Step 3.1.5

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

$0 - (- 1 \cdot 1) \sin (θ)$

Step 3.1.6

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

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Step 3.1.6.1

Multiply $- 1$ by $1$ .

$0 - - 1 \sin (θ)$

Step 3.1.6.2

Multiply $- 1$ by $- 1$ .

$0 + 1 \sin (θ)$

Step 3.1.7

Multiply $\sin (θ)$ by $1$ .

$0 + \sin (θ)$

Step 3.2

Add $0$ and $\sin (θ)$ .

$\sin (θ)$

Step 4

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

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