Trigonometry Examples

cos (\frac{π}{4} + \frac{π}{3})

$\cos (\frac{π}{4} + \frac{π}{3})$

Step 1

To write

\frac{π}{4}

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

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 2

To write

\frac{π}{3}

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

\frac{4}{4}

$\frac{4}{4}$ .

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

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

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}

$\frac{3}{3}$ .

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

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

Step 3.2

Multiply

4

$4$ by

3

$3$ .

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

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

Step 3.3

Multiply

\frac{π}{3}

$\frac{π}{3}$ by

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 3.4

Multiply

3

$3$ by

4

$4$ .

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Simplify the numerator.

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Step 5.1

Move

3

$3$ to the left of

π

$π$ .

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

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

Step 5.2

Move

4

$4$ to the left of

π

$π$ .

cos (\frac{3 π + 4 \cdot π}{12})

$\cos (\frac{3 π + 4 \cdot π}{12})$

Step 5.3

Add

3 π

$3 π$ and

4 π

$4 π$ .

cos (\frac{7 π}{12})

$\cos (\frac{7 π}{12})$

cos (\frac{7 π}{12})

$\cos (\frac{7 π}{12})$

Step 6

The exact value of

cos (\frac{7 π}{12})

$\cos (\frac{7 π}{12})$ 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

$2$ .

cos (\frac{\frac{7 π}{6}}{2})

$\cos (\frac{\frac{7 π}{6}}{2})$

Step 6.2

Apply the cosine half-angle identity

cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + cos (x)}{2}}

$\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

\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 cosine is negative 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

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 6.4.4

Combine the numerators over the common denominator.

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

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

Step 6.4.5

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 6.4.6

Multiply

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

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

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Step 6.4.6.1

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 6.4.6.2

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 6.4.7

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 6.4.8

Simplify the denominator.

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Step 6.4.8.1

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 6.4.8.2

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

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