Precalculus Examples

cos (75)

$\cos (75)$

Step 1

Split

75

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

cos (30 + 45)

$\cos (30 + 45)$

Step 2

Apply the sum of angles identity

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

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

cos (30) cos (45) - sin (30) sin (45)

$\cos (30) \cos (45) - \sin (30) \sin (45)$

Step 3

The exact value of

cos (30)

$\cos (30)$ is

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

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

\frac{\sqrt{3}}{2} cos (45) - sin (30) sin (45)

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

Step 4

The exact value of

cos (45)

$\cos (45)$ is

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

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

\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - sin (30) sin (45)

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)$

Step 5

The exact value of

sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} sin (45)

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45)$

Step 6

The exact value of

sin (45)

$\sin (45)$ is

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

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

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

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

Step 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 7.1

Simplify each term.

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Step 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 7.1.1.1

Multiply

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

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

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

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

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

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

Step 7.1.1.2

Combine using the product rule for radicals.

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

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

Step 7.1.1.3

Multiply

3

$3$ by

2

$2$ .

\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}

$\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

Step 7.1.1.4

Multiply

2

$2$ by

2

$2$ .

\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}

$\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$

Step 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 7.1.2.1

Multiply

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

$\frac{1}{2}$ .

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

Step 7.1.2.2

Multiply

$2$ by

$2$ .

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

Step 7.2

Combine the numerators over the common denominator.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.25881904 \dots$