Trigonometry Examples

$\tan (- 345)$

Step 1

Rewrite $- 345$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\tan (\frac{- 690}{2})$

Step 2

Apply the tangent half-angle identity.

$\pm \sqrt{\frac{1 - \cos (- 690)}{1 + \cos (- 690)}}$

Step 3

Change the $\pm$ to $+$ because tangent is positive in the first quadrant.

$\sqrt{\frac{1 - \cos (- 690)}{1 + \cos (- 690)}}$

Step 4

Simplify $\sqrt{\frac{1 - \cos (- 690)}{1 + \cos (- 690)}}$ .

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Step 4.1

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\sqrt{\frac{1 - \cos (30)}{1 + \cos (- 690)}}$

Step 4.2

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{1 + \cos (- 690)}}$

Step 4.3

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

$\sqrt{\frac{\frac{2}{2} - \frac{\sqrt{3}}{2}}{1 + \cos (- 690)}}$

Step 4.4

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 - \sqrt{3}}{2}}{1 + \cos (- 690)}}$

Step 4.5

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\sqrt{\frac{\frac{2 - \sqrt{3}}{2}}{1 + \cos (30)}}$

Step 4.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 4.7

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

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

Step 4.8

Combine the numerators over the common denominator.

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

Step 4.9

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.10

Cancel the common factor of $2$ .

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Step 4.10.1

Cancel the common factor.

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

Step 4.10.2

Rewrite the expression.

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

Step 4.11

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 4.12

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 4.13

Expand the denominator using the FOIL method.

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

Step 4.14

Simplify.

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

Step 4.15

Divide $2 - \sqrt{3}$ by $1$ .

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

Step 4.16

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

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Step 4.16.1

Apply the distributive property.

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

Step 4.16.2

Apply the distributive property.

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

Step 4.16.3

Apply the distributive property.

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

Step 4.17

Simplify and combine like terms.

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Step 4.17.1

Simplify each term.

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Step 4.17.1.1

Multiply $2$ by $2$ .

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

Step 4.17.1.2

Multiply $- 1$ by $2$ .

$\sqrt{4 - 2 \sqrt{3} - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3})}$

Step 4.17.1.3

Multiply $2$ by $- 1$ .

$\sqrt{4 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{3} (- \sqrt{3})}$

Step 4.17.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 4.17.1.4.1

Multiply $- 1$ by $- 1$ .

$\sqrt{4 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}$

Step 4.17.1.4.2

Multiply $\sqrt{3}$ by $1$ .

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

Step 4.17.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}$

Step 4.17.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.17.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1 + 1}}$

Step 4.17.1.4.6

Add $1$ and $1$ .

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

Step 4.17.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.17.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{4 - 2 \sqrt{3} - 2 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}$

Step 4.17.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}$

Step 4.17.1.5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.17.1.5.4

Cancel the common factor of $2$ .

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Step 4.17.1.5.4.1

Cancel the common factor.

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

Step 4.17.1.5.4.2

Rewrite the expression.

$\sqrt{4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{1}}$

Step 4.17.1.5.5

Evaluate the exponent.

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

Step 4.17.2

Add $4$ and $3$ .

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

Step 4.17.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.26794919 \dots$