Trigonometry Examples

$\sin (120) + \cos (45) + \tan (135) - \cos (30)$

Step 1

Simplify each term.

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Step 1.1

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

$\sin (60) + \cos (45) + \tan (135) - \cos (30)$

Step 1.2

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2} + \cos (45) + \tan (135) - \cos (30)$

Step 1.3

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

$\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} + \tan (135) - \cos (30)$

Step 1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} - \tan (45) - \cos (30)$

Step 1.5

The exact value of $\tan (45)$ is $1$ .

$\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} - 1 \cdot 1 - \cos (30)$

Step 1.6

Multiply $- 1$ by $1$ .

$\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} - 1 - \cos (30)$

Step 1.7

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} - 1 - \frac{\sqrt{3}}{2}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$- 1 + \frac{\sqrt{3} + \sqrt{2} - \sqrt{3}}{2}$

Step 2.2

Subtract $\sqrt{3}$ from $\sqrt{3}$ .

$- 1 + \frac{0 + \sqrt{2}}{2}$

Step 2.3

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

$- 1 + \frac{\sqrt{2}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 1 + \frac{\sqrt{2}}{2}$

Decimal Form:

$- 0.29289321 \dots$