Trigonometry Examples

\sin (\frac{π}{4} + \frac{π}{3})

$\sin (\frac{π}{4} + \frac{π}{3})$

Step 1

To write

\frac{π}{4}

$\frac{π}{4}$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

\sin (\frac{π}{4} \cdot \frac{3}{3} + \frac{π}{3})

$\sin (\frac{π}{4} \cdot \frac{3}{3} + \frac{π}{3})$

Step 2

To write

\frac{π}{3}

$\frac{π}{3}$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

\sin (\frac{π}{4} \cdot \frac{3}{3} + \frac{π}{3} \cdot \frac{4}{4})

$\sin (\frac{π}{4} \cdot \frac{3}{3} + \frac{π}{3} \cdot \frac{4}{4})$

Step 3

Write each expression with a common denominator of

12

$12$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{π}{4}

$\frac{π}{4}$ by

\frac{3}{3}

$\frac{3}{3}$ .

\sin (\frac{π \cdot 3}{4 \cdot 3} + \frac{π}{3} \cdot \frac{4}{4})

$\sin (\frac{π \cdot 3}{4 \cdot 3} + \frac{π}{3} \cdot \frac{4}{4})$

Step 3.2

Multiply

4

$4$ by

3

$3$ .

\sin (\frac{π \cdot 3}{12} + \frac{π}{3} \cdot \frac{4}{4})

$\sin (\frac{π \cdot 3}{12} + \frac{π}{3} \cdot \frac{4}{4})$

Step 3.3

Multiply

\frac{π}{3}

$\frac{π}{3}$ by

\frac{4}{4}

$\frac{4}{4}$ .

\sin (\frac{π \cdot 3}{12} + \frac{π \cdot 4}{3 \cdot 4})

$\sin (\frac{π \cdot 3}{12} + \frac{π \cdot 4}{3 \cdot 4})$

Step 3.4

Multiply

3

$3$ by

4

$4$ .

\sin (\frac{π \cdot 3}{12} + \frac{π \cdot 4}{12})

$\sin (\frac{π \cdot 3}{12} + \frac{π \cdot 4}{12})$

\sin (\frac{π \cdot 3}{12} + \frac{π \cdot 4}{12})

$\sin (\frac{π \cdot 3}{12} + \frac{π \cdot 4}{12})$

Step 4

Combine the numerators over the common denominator.

\sin (\frac{π \cdot 3 + π \cdot 4}{12})

$\sin (\frac{π \cdot 3 + π \cdot 4}{12})$

Step 5

Simplify the numerator.

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Step 5.1

Move

3

$3$ to the left of

π

$π$ .

\sin (\frac{3 \cdot π + π \cdot 4}{12})

$\sin (\frac{3 \cdot π + π \cdot 4}{12})$

Step 5.2

Move

4

$4$ to the left of

π

$π$ .

\sin (\frac{3 π + 4 \cdot π}{12})

$\sin (\frac{3 π + 4 \cdot π}{12})$

Step 5.3

Add

3 π

$3 π$ and

4 π

$4 π$ .

\sin (\frac{7 π}{12})

$\sin (\frac{7 π}{12})$

\sin (\frac{7 π}{12})

$\sin (\frac{7 π}{12})$

Step 6

The exact value of

\sin (\frac{7 π}{12})

$\sin (\frac{7 π}{12})$ is

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

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

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Step 6.1

Rewrite

\frac{7 π}{12}

$\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by

2

$2$ .

\sin (\frac{\frac{7 π}{6}}{2})

$\sin (\frac{\frac{7 π}{6}}{2})$

Step 6.2

Apply the sine half-angle identity.

\pm \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}

$\pm \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}$

Step 6.3

Change the

\pm

$\pm$ to

+

$+$ because sine is positive in the second quadrant.

\sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}

$\sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}$

Step 6.4

Simplify

\sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}

$\sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}$ .

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Step 6.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

\sqrt{\frac{1 - - \cos (\frac{π}{6})}{2}}

$\sqrt{\frac{1 - - \cos (\frac{π}{6})}{2}}$

Step 6.4.2

The exact value of

\cos (\frac{π}{6})

$\cos (\frac{π}{6})$ is

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

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

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

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

Step 6.4.3

Multiply

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

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

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Step 6.4.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 6.4.3.2

Multiply

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

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

1

$1$ .

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

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

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

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

Step 6.4.4

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 6.4.5

Combine the numerators over the common denominator.

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

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

Step 6.4.6

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 6.4.7

Multiply

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

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

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Step 6.4.7.1

Multiply

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

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

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 6.4.7.2

Multiply

2

$2$ by

2

$2$ .

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

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

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

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

Step 6.4.8

Rewrite

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

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

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

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

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

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

Step 6.4.9

Simplify the denominator.

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Step 6.4.9.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 6.4.9.2

Pull terms out from under the radical, assuming positive real numbers.

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