Trigonometry Examples

$\tan (300 + 60)$

Step 1

Use the sum formula for tangent to simplify the expression. The formula states that $\tan (A + B) = \frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}$ .

$\frac{\tan (300) + \tan (60)}{1 - \tan (300) \tan (60)}$

Step 2

Simplify the numerator.

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Step 2.1

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 fourth quadrant.

$\frac{- \tan (60) + \tan (60)}{1 - \tan (300) \tan (60)}$

Step 2.2

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\frac{- \sqrt{3} + \tan (60)}{1 - \tan (300) \tan (60)}$

Step 2.3

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\frac{- \sqrt{3} + \sqrt{3}}{1 - \tan (300) \tan (60)}$

Step 2.4

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

$\frac{0}{1 - \tan (300) \tan (60)}$

Step 3

Simplify the denominator.

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Step 3.1

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 fourth quadrant.

$\frac{0}{1 - - \tan (60) \tan (60)}$

Step 3.2

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\frac{0}{1 - - \sqrt{3} \tan (60)}$

Step 3.3

Multiply $- - \sqrt{3}$ .

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

$\frac{0}{1 + 1 \sqrt{3} \tan (60)}$

Step 3.3.2

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

$\frac{0}{1 + \sqrt{3} \tan (60)}$

Step 3.4

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\frac{0}{1 + \sqrt{3} \sqrt{3}}$

Step 3.5

Combine using the product rule for radicals.

$\frac{0}{1 + \sqrt{3 \cdot 3}}$

Step 3.6

Multiply $3$ by $3$ .

$\frac{0}{1 + \sqrt{9}}$

Step 3.7

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

$\frac{0}{1 + \sqrt{3^{2}}}$

Step 3.8

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

$\frac{0}{1 + 3}$

Step 3.9

Add $1$ and $3$ .

$\frac{0}{4}$

Step 4

Cancel the common factor of $0$ and $4$ .

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Step 4.1

Factor $4$ out of $0$ .

$\frac{4 (0)}{4}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 0}{4 \cdot 1}$

Step 4.2.2

Cancel the common factor.

$\frac{4 \cdot 0}{4 \cdot 1}$

Step 4.2.3

Rewrite the expression.

$\frac{0}{1}$

Step 4.2.4

Divide $0$ by $1$ .

$0$