Trigonometry Examples

$\tan (105)$

Step 1

The supplement of $\tan (105)$ is the angle that when added to $\tan (105)$ forms a straight angle ( $180 °$ ).

$180 ° - \tan (105)$

Step 2

Simplify each term.

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Step 2.1

The exact value of $\tan (105)$ is $- 2 - \sqrt{3}$ .

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Step 2.1.1

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

Step 2.1.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

Step 2.1.3

Apply the sum of angles identity.

Step 2.1.4

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

Step 2.1.5

The exact value of $\tan (45)$ is $1$ .

Step 2.1.6

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

Step 2.1.7

The exact value of $\tan (45)$ is $1$ .

Step 2.1.8

Simplify $- \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ .

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Step 2.1.8.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 2.1.8.1.1

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

Step 2.1.8.1.2

Combine.

Step 2.1.8.2

Apply the distributive property.

Step 2.1.8.3

Cancel the common factor of $3$ .

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Step 2.1.8.3.1

Cancel the common factor.

Step 2.1.8.3.2

Rewrite the expression.

Step 2.1.8.4

Multiply $3$ by $1$ .

Step 2.1.8.5

Simplify the denominator.

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Step 2.1.8.5.1

Multiply $3$ by $1$ .

Step 2.1.8.5.2

Multiply $- 1$ by $1$ .

Step 2.1.8.5.3

Cancel the common factor of $3$ .

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Step 2.1.8.5.3.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

Step 2.1.8.5.3.2

Cancel the common factor.

Step 2.1.8.5.3.3

Rewrite the expression.

Step 2.1.8.6

Multiply $\frac{\sqrt{3} + 3}{3 - \sqrt{3}}$ by $\frac{3 + \sqrt{3}}{3 + \sqrt{3}}$ .

Step 2.1.8.7

Multiply $\frac{\sqrt{3} + 3}{3 - \sqrt{3}}$ by $\frac{3 + \sqrt{3}}{3 + \sqrt{3}}$ .

Step 2.1.8.8

Expand the denominator using the FOIL method.

Step 2.1.8.9

Simplify.

$180 + \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{6 °}$

Step 2.1.8.10

Simplify the numerator.

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Step 2.1.8.10.1

Reorder terms.

$180 + \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6 °}$

Step 2.1.8.10.2

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

$180 + \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6 °}$

Step 2.1.8.10.3

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

$180 + \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6 °}$

Step 2.1.8.10.4

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

$180 + \frac{{(3 + \sqrt{3})}^{1 + 1}}{6 °}$

Step 2.1.8.10.5

Add $1$ and $1$ .

$180 + \frac{{(3 + \sqrt{3})}^{2}}{6 °}$

Step 2.1.8.11

Rewrite ${(3 + \sqrt{3})}^{2}$ as $(3 + \sqrt{3}) (3 + \sqrt{3})$ .

$180 + \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6 °}$

Step 2.1.8.12

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

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Step 2.1.8.12.1

Apply the distributive property.

$180 + \frac{3 (3 + \sqrt{3}) + \sqrt{3} (3 + \sqrt{3})}{6 °}$

Step 2.1.8.12.2

Apply the distributive property.

$180 + \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} (3 + \sqrt{3})}{6 °}$

Step 2.1.8.12.3

Apply the distributive property.

$180 + \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6 °}$

Step 2.1.8.13

Simplify and combine like terms.

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Step 2.1.8.13.1

Simplify each term.

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Step 2.1.8.13.1.1

Multiply $3$ by $3$ .

$180 + \frac{9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6 °}$

Step 2.1.8.13.1.2

Move $3$ to the left of $\sqrt{3}$ .

$180 + \frac{9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6 °}$

Step 2.1.8.13.1.3

Combine using the product rule for radicals.

$180 + \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3}}{6 °}$

Step 2.1.8.13.1.4

Multiply $3$ by $3$ .

$180 + \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6 °}$

Step 2.1.8.13.1.5

Rewrite $9$ as $3^{2}$ .

$180 + \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}}}{6 °}$

Step 2.1.8.13.1.6

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

$180 + \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + 3}{6 °}$

Step 2.1.8.13.2

Add $9$ and $3$ .

$180 + \frac{12 + 3 \sqrt{3} + 3 \sqrt{3}}{6 °}$

Step 2.1.8.13.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$180 + \frac{12 + 6 \sqrt{3}}{6 °}$

Step 2.1.8.14

Cancel the common factor of $12 + 6 \sqrt{3}$ and $6$ .

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Step 2.1.8.14.1

Factor $6$ out of $12$ .

$180 + \frac{6 \cdot 2 + 6 \sqrt{3}}{6 °}$

Step 2.1.8.14.2

Factor $6$ out of $6 \sqrt{3}$ .

$180 + \frac{6 \cdot 2 + 6 (\sqrt{3})}{6 °}$

Step 2.1.8.14.3

Factor $6$ out of $6 (2) + 6 (\sqrt{3})$ .

$180 + \frac{6 (2 + \sqrt{3})}{6 °}$

Step 2.1.8.14.4

Cancel the common factors.

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Step 2.1.8.14.4.1

Factor $6$ out of $6$ .

Step 2.1.8.14.4.2

Cancel the common factor.

Step 2.1.8.14.4.3

Rewrite the expression.

$180 + \frac{2 + \sqrt{3}}{1 °}$

Step 2.1.8.14.4.4

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

Step 2.1.8.15

Apply the distributive property.

Step 2.1.8.16

Multiply $- 1$ by $2$ .

Step 2.2

Apply the distributive property.

Step 2.3

Multiply $- 1$ by $- 2$ .

Step 2.4

Multiply $- - \sqrt{3}$ .

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Step 2.4.1

Multiply $- 1$ by $- 1$ .

Step 2.4.2

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

Step 3

Add $180$ and $2$ .