Trigonometry Examples

tan (\frac{5 π}{3} - \frac{π}{4})

$\tan (\frac{5 π}{3} - \frac{π}{4})$

Step 1

To write

$\frac{5 π}{3}$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 2

To write

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

$\frac{3}{3}$ .

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

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

$\frac{5 π}{3}$ by

$\frac{4}{4}$ .

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

Step 3.2

Multiply

$3$ by

$4$ .

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

Step 3.3

Multiply

$\frac{π}{4}$ by

$\frac{3}{3}$ .

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

Step 3.4

Multiply

$4$ by

$3$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply

$4$ by

$5$ .

$\tan (\frac{20 π - π \cdot 3}{12})$

Step 5.2

Multiply

$3$ by

$- 1$ .

$\tan (\frac{20 π - 3 π}{12})$

Step 5.3

Subtract

$3 π$ from

$20 π$ .

$\tan (\frac{17 π}{12})$

Step 6

The exact value of

$\tan (\frac{17 π}{12})$ is

$\sqrt{7 + 4 \sqrt{3}}$ .

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Step 6.1

Rewrite

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

$2$ .

$\tan (\frac{\frac{17 π}{6}}{2})$

Step 6.2

Apply the tangent half-angle identity.

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

Step 6.3

Change the

$\pm$ to

$+$ because tangent is positive in the third quadrant.

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

Step 6.4

Simplify

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

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Step 6.4.1

Subtract full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$\sqrt{\frac{1 - \cos (\frac{5 π}{6})}{1 + \cos (\frac{17 π}{6})}}$

Step 6.4.2

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.

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

Step 6.4.3

The exact value of

$\cos (\frac{π}{6})$ is

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

$\sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{1 + \cos (\frac{17 π}{6})}}$

Step 6.4.4

Multiply

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

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Step 6.4.4.1

Multiply

$- 1$ by

$- 1$ .

$\sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{1 + \cos (\frac{17 π}{6})}}$

Step 6.4.4.2

Multiply

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

$1$ .

$\sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{17 π}{6})}}$

Step 6.4.5

Write

$1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{1 + \cos (\frac{17 π}{6})}}$

Step 6.4.6

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 + \cos (\frac{17 π}{6})}}$

Step 6.4.7

Subtract full rotations of

$2 π$ until the angle is greater than or equal to

$0$ and less than

$2 π$ .

$\sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 + \cos (\frac{5 π}{6})}}$

Step 6.4.8

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.

$\sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{1 - \cos (\frac{π}{6})}}$

Step 6.4.9

The exact value of

$\cos (\frac{π}{6})$ is

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

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

Step 6.4.10

Write

$1$ as a fraction with a common denominator.

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

Step 6.4.11

Combine the numerators over the common denominator.

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

Step 6.4.12

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.4.13

Cancel the common factor of

$2$ .

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Step 6.4.13.1

Cancel the common factor.

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

Step 6.4.13.2

Rewrite the expression.

$\sqrt{(2 + \sqrt{3}) \frac{1}{2 - \sqrt{3}}}$

Step 6.4.14

Multiply

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

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

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

Step 6.4.15

Multiply

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

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

$\sqrt{(2 + \sqrt{3}) \frac{2 + \sqrt{3}}{(2 - \sqrt{3}) (2 + \sqrt{3})}}$

Step 6.4.16

Expand the denominator using the FOIL method.

$\sqrt{(2 + \sqrt{3}) \frac{2 + \sqrt{3}}{4 + 2 \sqrt{3} - 2 \sqrt{3} - {\sqrt{3}}^{2}}}$

Step 6.4.17

Simplify.

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

Step 6.4.18

Divide

$2 + \sqrt{3}$ by

$1$ .

$\sqrt{(2 + \sqrt{3}) (2 + \sqrt{3})}$

Step 6.4.19

Expand

$(2 + \sqrt{3}) (2 + \sqrt{3})$ using the FOIL Method.

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Step 6.4.19.1

Apply the distributive property.

$\sqrt{2 (2 + \sqrt{3}) + \sqrt{3} (2 + \sqrt{3})}$

Step 6.4.19.2

Apply the distributive property.

$\sqrt{2 \cdot 2 + 2 \sqrt{3} + \sqrt{3} (2 + \sqrt{3})}$

Step 6.4.19.3

Apply the distributive property.

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

Step 6.4.20

Simplify and combine like terms.

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Step 6.4.20.1

Simplify each term.

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Step 6.4.20.1.1

Multiply

$2$ by

$2$ .

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

Step 6.4.20.1.2

Move

$2$ to the left of

$\sqrt{3}$ .

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

Step 6.4.20.1.3

Combine using the product rule for radicals.

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

Step 6.4.20.1.4

Multiply

$3$ by

$3$ .

$\sqrt{4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{9}}$

Step 6.4.20.1.5

Rewrite

$9$ as

$3^{2}$ .

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

Step 6.4.20.1.6

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

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

Step 6.4.20.2

Add

$4$ and

$3$ .

$\sqrt{7 + 2 \sqrt{3} + 2 \sqrt{3}}$

Step 6.4.20.3

Add

$2 \sqrt{3}$ and

$2 \sqrt{3}$ .

$\sqrt{7 + 4 \sqrt{3}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt{7 + 4 \sqrt{3}}$

Decimal Form:

$3.73205080 \dots$