Trigonometry Examples

$\cos (\frac{2 π}{3} - \frac{5 π}{4})$

Step 1

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\cos (\frac{2 π}{3} \cdot \frac{4}{4} - \frac{5 π}{4})$

Step 2

To write $- \frac{5 π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\cos (\frac{2 π}{3} \cdot \frac{4}{4} - \frac{5 π}{4} \cdot \frac{3}{3})$

Step 3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2 π}{3}$ by $\frac{4}{4}$ .

$\cos (\frac{2 π \cdot 4}{3 \cdot 4} - \frac{5 π}{4} \cdot \frac{3}{3})$

Step 3.2

Multiply $3$ by $4$ .

$\cos (\frac{2 π \cdot 4}{12} - \frac{5 π}{4} \cdot \frac{3}{3})$

Step 3.3

Multiply $\frac{5 π}{4}$ by $\frac{3}{3}$ .

$\cos (\frac{2 π \cdot 4}{12} - \frac{5 π \cdot 3}{4 \cdot 3})$

Step 3.4

Multiply $4$ by $3$ .

$\cos (\frac{2 π \cdot 4}{12} - \frac{5 π \cdot 3}{12})$

Step 4

Combine the numerators over the common denominator.

$\cos (\frac{2 π \cdot 4 - 5 π \cdot 3}{12})$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $4$ by $2$ .

$\cos (\frac{8 π - 5 π \cdot 3}{12})$

Step 5.2

Multiply $3$ by $- 5$ .

$\cos (\frac{8 π - 15 π}{12})$

Step 5.3

Subtract $15 π$ from $8 π$ .

$\cos (\frac{- 7 π}{12})$

Step 6

Move the negative in front of the fraction.

$\cos (- \frac{7 π}{12})$

Step 7

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\cos (\frac{17 π}{12})$

Step 8

The exact value of $\cos (\frac{17 π}{12})$ is $- \frac{\sqrt{2 - \sqrt{3}}}{2}$ .

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Step 8.1

Rewrite $\frac{17 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\cos (\frac{\frac{17 π}{6}}{2})$

Step 8.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$\pm \sqrt{\frac{1 + \cos (\frac{17 π}{6})}{2}}$

Step 8.3

Change the $\pm$ to $-$ because cosine is negative in the third quadrant.

$- \sqrt{\frac{1 + \cos (\frac{17 π}{6})}{2}}$

Step 8.4

Simplify $- \sqrt{\frac{1 + \cos (\frac{17 π}{6})}{2}}$ .

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Step 8.4.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- \sqrt{\frac{1 + \cos (\frac{5 π}{6})}{2}}$

Step 8.4.2

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 second quadrant.

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

Step 8.4.3

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 8.4.4

Write $1$ as a fraction with a common denominator.

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

Step 8.4.5

Combine the numerators over the common denominator.

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

Step 8.4.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.4.7

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

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Step 8.4.7.1

Multiply $\frac{2 - \sqrt{3}}{2}$ by $\frac{1}{2}$ .

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

Step 8.4.7.2

Multiply $2$ by $2$ .

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

Step 8.4.8

Rewrite $\sqrt{\frac{2 - \sqrt{3}}{4}}$ as $\frac{\sqrt{2 - \sqrt{3}}}{\sqrt{4}}$ .

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

Step 8.4.9

Simplify the denominator.

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Step 8.4.9.1

Rewrite $4$ as $2^{2}$ .

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

Step 8.4.9.2

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

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.25881904 \dots$