Trigonometry Examples

$\cos (x + 90)$

Step 1

Double negation

$\cos (x - (- 90))$

Step 2

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

$\cos (x) \cos (- 90) + \sin (x) \sin (- 90)$

Step 3

Simplify terms.

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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. Make the expression negative because cosine is negative in the third quadrant.

$\cos (x) (- \cos (90)) + \sin (x) \sin (- 90)$

Step 3.1.2

The exact value of $\cos (90)$ is $0$ .

$\cos (x) (- 0) + \sin (x) \sin (- 90)$

Step 3.1.3

Multiply $- 1$ by $0$ .

$\cos (x) \cdot 0 + \sin (x) \sin (- 90)$

Step 3.1.4

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

$0 + \sin (x) \sin (- 90)$

Step 3.1.5

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 third quadrant.

$0 + \sin (x) (- \sin (90))$

Step 3.1.6

The exact value of $\sin (90)$ is $1$ .

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

Step 3.1.7

Multiply $- 1$ by $1$ .

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

Step 3.1.8

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

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

Step 3.1.9

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

$0 - \sin (x)$

Step 3.2

Subtract $\sin (x)$ from $0$ .

$- \sin (x)$