Trigonometry Examples

$\cos (45 + x)$

Step 1

Rearrange terms.

$\cos (x + 45)$

Step 2

Double negation

$\cos (x - (- 45))$

Step 3

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

$\cos (x) \cos (- 45) + \sin (x) \sin (- 45)$

Step 4

Simplify each term.

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Step 4.1

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

$\cos (x) \cos (45) + \sin (x) \sin (- 45)$

Step 4.2

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\cos (x) \frac{\sqrt{2}}{2} + \sin (x) \sin (- 45)$

Step 4.3

Combine $\cos (x)$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\cos (x) \sqrt{2}}{2} + \sin (x) \sin (- 45)$

Step 4.4

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.

$\frac{\cos (x) \sqrt{2}}{2} + \sin (x) (- \sin (45))$

Step 4.5

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

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

Step 4.6

Combine $\sin (x)$ and $\frac{\sqrt{2}}{2}$ .

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