Trigonometry Examples

\sin (75)

$\sin (75)$

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

$30 + 45$ .

\sin (30 + 45)

$\sin (30 + 45)$

Step 2

Use the sum formula for sine to simplify the expression. The formula states that

\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)

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

\sin (30) \cos (45) + \cos (30) \sin (45)

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

Step 3

Remove parentheses.

\sin (30) \cos (45) + \cos (30) \sin (45)

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

Step 4

Simplify each term.

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Step 4.1

The exact value of

\sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1}{2} \cos (45) + \cos (30) \sin (45)

$\frac{1}{2} \cos (45) + \cos (30) \sin (45)$

Step 4.2

The exact value of

\cos (45)

$\cos (45)$ is

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

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

\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)

$\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)$

Step 4.3

Multiply

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

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

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Step 4.3.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

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

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

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

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

Step 4.3.2

Multiply

2

$2$ by

2

$2$ .

\frac{\sqrt{2}}{4} + \cos (30) \sin (45)

$\frac{\sqrt{2}}{4} + \cos (30) \sin (45)$

\frac{\sqrt{2}}{4} + \cos (30) \sin (45)

$\frac{\sqrt{2}}{4} + \cos (30) \sin (45)$

Step 4.4

The exact value of

\cos (30)

$\cos (30)$ is

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

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

\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \sin (45)

$\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \sin (45)$

Step 4.5

The exact value of

\sin (45)

$\sin (45)$ is

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

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

\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$

Step 4.6

Multiply

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

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

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Step 4.6.1

Multiply

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

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

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

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

\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}

$\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 4.6.2

Combine using the product rule for radicals.

\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}

$\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}$

Step 4.6.3

Multiply

3

$3$ by

2

$2$ .

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

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

Step 4.6.4

Multiply

2

$2$ by

2

$2$ .

\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}

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

Step 5

Combine the numerators over the common denominator.

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

0.96592582 \dots

$0.96592582 \dots$

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