Trigonometry Examples

sin (\frac{3 π}{2} + x)

$\sin (\frac{3 π}{2} + x)$

Step 1

Apply the sum of angles identity.

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

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

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)

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

Step 2.1.2

The exact value of

sin (\frac{π}{2})

$\sin (\frac{π}{2})$ is

1

$1$ .

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

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

Step 2.1.3

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 2.1.4

Rewrite

- 1 cos (x)

$- 1 \cos (x)$ as

- cos (x)

$- \cos (x)$ .

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

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

Step 2.1.5

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

- cos (x) + cos (\frac{π}{2}) sin (x)

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

Step 2.1.6

The exact value of

cos (\frac{π}{2})

$\cos (\frac{π}{2})$ is

0

$0$ .

- cos (x) + 0 sin (x)

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

Step 2.1.7

Multiply

0

$0$ by

sin (x)

$\sin (x)$ .

- cos (x) + 0

$- \cos (x) + 0$

- cos (x) + 0

$- \cos (x) + 0$

Step 2.2

Add

- cos (x)

$- \cos (x)$ and

0

$0$ .

- cos (x)

$- \cos (x)$

- cos (x)

$- \cos (x)$