Trigonometry Examples

$\cot (- 3) = \tan (x)$

Step 1

Rewrite the equation as $\tan (x) = \cot (- 3)$ .

$\tan (x) = \cot (- 3)$

Step 2

Evaluate $\cot (- 3)$ .

$\tan (x) = - 19.08113668$

Step 3

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (- 19.08113668)$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arctan (- 19.08113668)$ .

$x = - 87$

Step 5

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the third quadrant.

$x = - 87 - 180$

Step 6

Simplify the expression to find the second solution.

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Step 6.1

Add $360 °$ to $- 87 - 180 °$ .

$x = - 87 - 180 ° + 360 °$

Step 6.2

The resulting angle of $93 °$ is positive and coterminal with $- 87 - 180$ .

$x = 93 °$

Step 7

Find the period of $\tan (x)$ .

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Step 7.1

The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 7.2

Replace $b$ with $1$ in the formula for period.

$\frac{180}{| 1 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{180}{1}$

Step 7.4

Divide $180$ by $1$ .

$180$

Step 8

Add $180$ to every negative angle to get positive angles.

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Step 8.1

Add $180$ to $- 87$ to find the positive angle.

$- 87 + 180$

Step 8.2

Subtract $87$ from $180$ .

$93$

Step 8.3

List the new angles.

$x = 93$

Step 9

The period of the $\tan (x)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$x = 93 + 180 n, 93 + 180 n$ , for any integer $n$

Step 10

Consolidate the answers.

$x = 93 + 180 n$ , for any integer $n$