Precalculus Examples

$\tan (x) = \frac{12}{5}$ , $π < x < \frac{3 π}{2}$

Step 1

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

$x = \arctan (\frac{12}{5})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arctan (\frac{12}{5})$ .

$x = 1.1760052$

Step 3

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

$x = (3.14159265) + 1.1760052$

Step 4

Solve for $x$ .

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Step 4.1

Remove parentheses.

$x = 3.14159265 + 1.1760052$

Step 4.2

Remove parentheses.

$x = (3.14159265) + 1.1760052$

Step 4.3

Add $3.14159265$ and $1.1760052$ .

$x = 4.31759786$

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

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

$x = 1.1760052 + π n, 4.31759786 + π n$ , for any integer $n$

Step 7

Consolidate $1.1760052 + π n$ and $4.31759786 + π n$ to $1.1760052 + π n$ .

$x = 1.1760052 + π n$ , for any integer $n$

Step 8

Plug in $1$ for $n$ and simplify to see if the solution is contained in $(π, \frac{3 π}{2})$ .

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Step 8.1

Plug in $1$ for $n$ .

$1.1760052 + π (1)$

Step 8.2

Simplify.

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Step 8.2.1

Multiply $π$ by $1$ .

$1.1760052 + π$

Step 8.2.2

Replace with decimal approximation.

$1.1760052 + 3.14159265$

Step 8.2.3

Add $1.1760052$ and $3.14159265$ .

$4.31759786$

Step 8.3

The interval $(π, \frac{3 π}{2})$ contains $4.31759786$ .

$x = 4.31759786$