Trigonometry Examples

$\tan (x) = - 10$

Step 1

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

$x = \arctan (- 10)$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arctan (- 10)$ .

$x = - 1.47112767$

Step 3

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

$x = - 1.47112767 - (3.14159265)$

Step 4

Simplify the expression to find the second solution.

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Step 4.1

Add $2 π$ to $- 1.47112767 - (3.14159265)$ .

$x = - 1.47112767 - (3.14159265) + 2 π$

Step 4.2

The resulting angle of $1.67046497$ is positive and coterminal with $- 1.47112767 - (3.14159265)$ .

$x = 1.67046497$

Step 5

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

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Step 5.1

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

$\frac{π}{| b |}$

Step 5.2

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

$\frac{π}{| 1 |}$

Step 5.3

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

$\frac{π}{1}$

Step 5.4

Divide $π$ by $1$ .

$π$

Step 6

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

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Step 6.1

Add $π$ to $- 1.47112767$ to find the positive angle.

$- 1.47112767 + π$

Step 6.2

Replace with decimal approximation.

$3.14159265 - 1.47112767$

Step 6.3

Subtract $1.47112767$ from $3.14159265$ .

$1.67046497$

Step 6.4

List the new angles.

$x = 1.67046497$

Step 7

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

$x = 1.67046497 + π n, 1.67046497 + π n$ , for any integer $n$

Step 8

Consolidate $1.67046497 + π n$ and $1.67046497 + π n$ to $1.67046497 + π n$ .

$x = 1.67046497 + π n$ , for any integer $n$