Trigonometry Examples

$\tan (- 135)$

Step 1

The angle $- 135$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

$\tan (- 135 + 0)$

Step 2

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

$\frac{\tan (0) - \tan (135)}{1 + \tan (0) \tan (135)}$

Step 3

Remove parentheses.

$\frac{\tan (0) - \tan (135)}{1 + \tan (0) \tan (135)}$

Step 4

Simplify the numerator.

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Step 4.1

The exact value of $\tan (0)$ is $0$ .

$\frac{0 - \tan (135)}{1 + \tan (0) \tan (135)}$

Step 4.2

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.

$\frac{0 - - \tan (45)}{1 + \tan (0) \tan (135)}$

Step 4.3

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

$\frac{0 - (- 1 \cdot 1)}{1 + \tan (0) \tan (135)}$

Step 4.4

Multiply $- (- 1 \cdot 1)$ .

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Step 4.4.1

Multiply $- 1$ by $1$ .

$\frac{0 - - 1}{1 + \tan (0) \tan (135)}$

Step 4.4.2

Multiply $- 1$ by $- 1$ .

$\frac{0 + 1}{1 + \tan (0) \tan (135)}$

Step 4.5

Add $0$ and $1$ .

$\frac{1}{1 + \tan (0) \tan (135)}$

Step 5

Simplify the denominator.

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Step 5.1

The exact value of $\tan (0)$ is $0$ .

$\frac{1}{1 + 0 \tan (135)}$

Step 5.2

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.

$\frac{1}{1 + 0 (- \tan (45))}$

Step 5.3

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

$\frac{1}{1 + 0 (- 1 \cdot 1)}$

Step 5.4

Multiply $0 (- 1 \cdot 1)$ .

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Step 5.4.1

Multiply $- 1$ by $1$ .

$\frac{1}{1 + 0 \cdot - 1}$

Step 5.4.2

Multiply $0$ by $- 1$ .

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

Step 5.5

Add $1$ and $0$ .

$\frac{1}{1}$

Step 6

Cancel the common factor of $1$ .

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Step 6.1

Cancel the common factor.

$\frac{1}{1}$

Step 6.2

Rewrite the expression.

$1$