Trigonometry Examples

cos (75)

$\cos (75)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case,

75

$75$ can be split into

30 + 45

$30 + 45$ .

cos (30 + 45)

$\cos (30 + 45)$

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))

$\cos (A + B) = - (\cos (A) \cos (B) + \sin (A) \sin (B))$ .

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

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

Step 3

Remove parentheses.

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

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

Step 4

Simplify each term.

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Step 4.1

The exact value of

cos (45)

$\cos (45)$ is

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

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

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

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

Step 4.2

The exact value of

cos (30)

$\cos (30)$ is

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

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

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

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

Step 4.3

Multiply

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

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

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Step 4.3.1

Multiply

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

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

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

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

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

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

Step 4.3.2

Combine using the product rule for radicals.

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

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

Step 4.3.3

Multiply

2

$2$ by

3

$3$ .

\frac{\sqrt{6}}{2 \cdot 2} - sin (45) \cdot sin (30)

$\frac{\sqrt{6}}{2 \cdot 2} - \sin (45) \cdot \sin (30)$

Step 4.3.4

Multiply

2

$2$ by

2

$2$ .

\frac{\sqrt{6}}{4} - sin (45) \cdot sin (30)

$\frac{\sqrt{6}}{4} - \sin (45) \cdot \sin (30)$

\frac{\sqrt{6}}{4} - sin (45) \cdot sin (30)

$\frac{\sqrt{6}}{4} - \sin (45) \cdot \sin (30)$

Step 4.4

The exact value of

sin (45)

$\sin (45)$ is

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

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

\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot sin (30)

$\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \sin (30)$

Step 4.5

The exact value of

sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 4.6

Multiply

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

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

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Step 4.6.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

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

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

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

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

Step 4.6.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

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

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

Step 5

Combine the numerators over the common denominator.

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.25881904 \dots

$0.25881904 \dots$

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