Trigonometry Examples

$\cos (105)$

Step 1

The supplement of $\cos (105)$ is the angle that when added to $\cos (105)$ forms a straight angle ( $180 °$ ).

$180 ° - \cos (105)$

Step 2

Simplify each term.

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Step 2.1

The exact value of $\cos (105)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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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 cosine is negative in the second quadrant.

Step 2.1.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

Step 2.1.3

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

Step 2.1.4

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

Step 2.1.5

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

Step 2.1.6

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

Step 2.1.7

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

Step 2.1.8

Simplify $- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$ .

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Step 2.1.8.1

Simplify each term.

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Step 2.1.8.1.1

Multiply $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 2.1.8.1.1.1

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{2}}{2}$ .

Step 2.1.8.1.1.2

Combine using the product rule for radicals.

Step 2.1.8.1.1.3

Multiply $3$ by $2$ .

Step 2.1.8.1.1.4

Multiply $2$ by $2$ .

Step 2.1.8.1.2

Multiply $- \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 2.1.8.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

Step 2.1.8.1.2.2

Multiply $2$ by $2$ .

Step 2.1.8.2

Combine the numerators over the common denominator.

$180 + \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 2.2

Multiply $- - \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.2.1

Multiply $- 1$ by $- 1$ .

Step 2.2.2

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $1$ .

$180 + \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 3

To write $180$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$180 \cdot \frac{4}{4} + \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 4

Combine $180$ and $\frac{4}{4}$ .

$\frac{180 \cdot 4}{4} + \frac{\sqrt{6} - \sqrt{2}}{4 °}$

Step 5

Simplify the expression.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{180 \cdot 4 + \sqrt{6} - \sqrt{2}}{4 °}$

Step 5.2

Multiply $180$ by $4$ .

$\frac{720 + \sqrt{6} - \sqrt{2}}{4 °}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{720 + \sqrt{6} - \sqrt{2}}{4 °}$

Decimal Form:

$180.25881904 \dots$