Trigonometry Examples

$\cos (135)$

Step 1

The angle $135$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

$\cos (135 + 0)$

Step 2

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

$\cos (0) \cdot \cos (135) - \sin (0) \cdot \sin (135)$

Step 3

Remove parentheses.

$\cos (0) \cdot \cos (135) - \sin (0) \cdot \sin (135)$

Step 4

Simplify each term.

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Step 4.1

The exact value of $\cos (0)$ is $1$ .

$1 \cdot \cos (135) - \sin (0) \cdot \sin (135)$

Step 4.2

Multiply $\cos (135)$ by $1$ .

$\cos (135) - \sin (0) \cdot \sin (135)$

Step 4.3

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 second quadrant.

$- \cos (45) - \sin (0) \cdot \sin (135)$

Step 4.4

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

$- \frac{\sqrt{2}}{2} - \sin (0) \cdot \sin (135)$

Step 4.5

The exact value of $\sin (0)$ is $0$ .

$- \frac{\sqrt{2}}{2} - 0 \cdot \sin (135)$

Step 4.6

Multiply $- 1$ by $0$ .

$- \frac{\sqrt{2}}{2} + 0 \cdot \sin (135)$

Step 4.7

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

$- \frac{\sqrt{2}}{2} + 0 \cdot \sin (45)$

Step 4.8

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

$- \frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2}$

Step 4.9

Multiply $0$ by $\frac{\sqrt{2}}{2}$ .

$- \frac{\sqrt{2}}{2} + 0$

Step 5

Add $- \frac{\sqrt{2}}{2}$ and $0$ .

$- \frac{\sqrt{2}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{2}}{2}$

Decimal Form:

$- 0.70710678 \dots$

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