Trigonometry Examples

$\cos (\frac{3 π}{2} + x)$

Step 1

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

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

Step 2

Remove parentheses.

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

Step 3

Simplify each term.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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.

$0 - \sin (x) \cdot (- \sin (\frac{π}{2}))$

Step 3.5

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

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

Step 3.6

Multiply $- 1$ by $1$ .

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

Step 3.7

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

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Step 3.7.1

Multiply $- 1$ by $- 1$ .

$0 + 1 \sin (x)$

Step 3.7.2

Multiply $\sin (x)$ by $1$ .

$0 + \sin (x)$

Step 4

Add $0$ and $\sin (x)$ .

$\sin (x)$