Trigonometry Examples

\sin (75)

$\sin (75)$

Step 1

Split

75

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

\sin (30 + 45)

$\sin (30 + 45)$

Step 2

Apply the sum of angles identity.

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

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

Step 3

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

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 5

The exact value of

\cos (30)

$\cos (30)$ is

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

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

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

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

Step 6

The exact value of

\sin (45)

$\sin (45)$ is

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

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

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

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

Step 7

Simplify

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

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

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply

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

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

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Step 7.1.1.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

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

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

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

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

Step 7.1.1.2

Multiply

2

$2$ by

2

$2$ .

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

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

\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 7.1.2

Multiply

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

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

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Step 7.1.2.1

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 7.1.2.2

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 7.1.2.3

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 7.1.2.4

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 7.2

Combine the numerators over the common denominator.

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

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

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

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

Step 8

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

$0.96592582 \dots$