Trigonometry Examples

sin (75)

$\sin (75)$

Step 1

Convert the radian measure to degrees.

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Step 1.1

To convert radians to degrees, multiply by

\frac{180}{π}

$\frac{180}{π}$ , since a full circle is

360 °

$360 °$ or

2 π

$2 π$ radians.

(sin (75)) \cdot \frac{180 °}{π}

$(\sin (75)) \cdot \frac{180 °}{π}$

Step 1.2

The exact value of

sin (75)

$\sin (75)$ is

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

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

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Step 1.2.1

Split

75

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

sin (30 + 45) \cdot \frac{180}{π}

$\sin (30 + 45) \cdot \frac{180}{π}$

Step 1.2.2

Apply the sum of angles identity.

(sin (30) cos (45) + cos (30) sin (45)) \cdot \frac{180}{π}

$(\sin (30) \cos (45) + \cos (30) \sin (45)) \cdot \frac{180}{π}$

Step 1.2.3

The exact value of

sin (30)

$\sin (30)$ is

\frac{1}{2}

$\frac{1}{2}$ .

(\frac{1}{2} cos (45) + cos (30) sin (45)) \cdot \frac{180}{π}

$(\frac{1}{2} \cos (45) + \cos (30) \sin (45)) \cdot \frac{180}{π}$

Step 1.2.4

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)) \cdot \frac{180}{π}

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) \cdot \frac{180}{π}$

Step 1.2.5

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)) \cdot \frac{180}{π}

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)) \cdot \frac{180}{π}$

Step 1.2.6

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}) \cdot \frac{180}{π}

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cdot \frac{180}{π}$

Step 1.2.7

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 1.2.7.1

Simplify each term.

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Step 1.2.7.1.1

Multiply

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

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

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Step 1.2.7.1.1.1

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}) \cdot \frac{180}{π}

$(\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cdot \frac{180}{π}$

Step 1.2.7.1.1.2

Multiply

$2$ by

$2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cdot \frac{180}{π}$

Step 1.2.7.1.2

Multiply

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

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Step 1.2.7.1.2.1

Multiply

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

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

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) \cdot \frac{180}{π}$

Step 1.2.7.1.2.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) \cdot \frac{180}{π}$

Step 1.2.7.1.2.3

Multiply

$3$ by

$2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}) \cdot \frac{180}{π}$

Step 1.2.7.1.2.4

Multiply

$2$ by

$2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}) \cdot \frac{180}{π}$

Step 1.2.7.2

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{180}{π}$

Step 1.3

Simplify terms.

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Step 1.3.1

Cancel the common factor of

$4$ .

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Step 1.3.1.1

Factor

$4$ out of

$180$ .

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{4 (45)}{π}$

Step 1.3.1.2

Cancel the common factor.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{4 \cdot 45}{π}$

Step 1.3.1.3

Rewrite the expression.

$(\sqrt{2} + \sqrt{6}) \cdot \frac{45}{π}$

Step 1.3.2

Apply the distributive property.

$\sqrt{2} \frac{45}{π} + \sqrt{6} \frac{45}{π}$

Step 1.3.3

Combine

$\sqrt{2}$ and

$\frac{45}{π}$ .

$\frac{\sqrt{2} \cdot 45}{π} + \sqrt{6} \frac{45}{π}$

Step 1.3.4

Combine

$\sqrt{6}$ and

$\frac{45}{π}$ .

$\frac{\sqrt{2} \cdot 45}{π} + \frac{\sqrt{6} \cdot 45}{π}$

Step 1.4

Simplify each term.

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Step 1.4.1

Move

$45$ to the left of

$\sqrt{2}$ .

$\frac{45 \sqrt{2}}{π} + \frac{\sqrt{6} \cdot 45}{π}$

Step 1.4.2

Move

$45$ to the left of

$\sqrt{6}$ .

$\frac{45 \sqrt{2}}{π} + \frac{45 \sqrt{6}}{π}$

Step 1.5

$π$ is approximately equal to

$3.14159265$ .

$\frac{45 \sqrt{2}}{3.14159265} + \frac{45 \sqrt{6}}{3.14159265}$

Step 1.6

Convert to a decimal.

$55.34347316 °$

Step 2

The angle is in the first quadrant.

Quadrant

$1$

Step 3