Trigonometry Examples

\cos (75)

$\cos (75)$

Step 1

The supplement of

\cos (75)

$\cos (75)$ is the angle that when added to

\cos (75)

$\cos (75)$ forms a straight angle (

180 °

$180 °$ ).

180 ° - \cos (75)

$180 ° - \cos (75)$

Step 2

The exact value of

\cos (75)

$\cos (75)$ is

\frac{\sqrt{6} - \sqrt{2}}{4}

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

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Step 2.1

Split

75

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

Step 2.2

Apply the sum of angles identity

\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)

$\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

Step 2.3

The exact value of

\cos (30)

$\cos (30)$ is

\frac{\sqrt{3}}{2}

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

Step 2.4

The exact value of

\cos (45)

$\cos (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

Step 2.5

The exact value of

\sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

Step 2.6

The exact value of

\sin (45)

$\sin (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

Step 2.7

Simplify

\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}

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

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Multiply

\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}

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

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Step 2.7.1.1.1

Multiply

\frac{\sqrt{3}}{2}

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

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

Step 2.7.1.1.2

Combine using the product rule for radicals.

Step 2.7.1.1.3

Multiply

3

$3$ by

2

$2$ .

Step 2.7.1.1.4

Multiply

2

$2$ by

2

$2$ .

Step 2.7.1.2

Multiply

- \frac{1}{2} \cdot \frac{\sqrt{2}}{2}

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

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Step 2.7.1.2.1

Multiply

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ by

\frac{1}{2}

$\frac{1}{2}$ .

Step 2.7.1.2.2

Multiply

2

$2$ by

2

$2$ .

Step 2.7.2

Combine the numerators over the common denominator.

180 - \frac{\sqrt{6} - \sqrt{2}}{4 °}

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

180 - \frac{\sqrt{6} - \sqrt{2}}{4 °}

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

180 - \frac{\sqrt{6} - \sqrt{2}}{4 °}

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

Step 3

To write

180

$180$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 4

Combine fractions.

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Step 4.1

Combine

180

$180$ and

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 4.2

Combine the numerators over the common denominator.

\frac{180 \cdot 4 - (\sqrt{6} - \sqrt{2})}{4 °}

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

\frac{180 \cdot 4 - (\sqrt{6} - \sqrt{2})}{4 °}

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply

180

$180$ by

4

$4$ .

\frac{720 - (\sqrt{6} - \sqrt{2})}{4 °}

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

Step 5.2

Apply the distributive property.

\frac{720 - \sqrt{6} + \sqrt{2}}{4 °}

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

Step 5.3

Multiply

- - \sqrt{2}

$- - \sqrt{2}$ .

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Step 5.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{720 - \sqrt{6} + 1 \sqrt{2}}{4 °}

$\frac{720 - \sqrt{6} + 1 \sqrt{2}}{4 °}$

Step 5.3.2

Multiply

\sqrt{2}

$\sqrt{2}$ by

1

$1$ .

\frac{720 - \sqrt{6} + \sqrt{2}}{4 °}

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

\frac{720 - \sqrt{6} + \sqrt{2}}{4 °}

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

\frac{720 - \sqrt{6} + \sqrt{2}}{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 °}

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

Decimal Form:

179.74118095 \dots

$179.74118095 \dots$