Trigonometry Examples

$f (x) = \tan (3 x)$ , $f (x) = 0.5$

Step 1

Substitute $0.5$ for $f (x)$ .

$0.5 = \tan (3 x)$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $\tan (3 x) = 0.5$ .

$\tan (3 x) = 0.5$

Step 2.2

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

$3 x = \arctan (0.5)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Evaluate $\arctan (0.5)$ .

$3 x = 0.4636476$

Step 2.4

Divide each term in $3 x = 0.4636476$ by $3$ and simplify.

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Step 2.4.1

Divide each term in $3 x = 0.4636476$ by $3$ .

$\frac{3 x}{3} = \frac{0.4636476}{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0.4636476}{3}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.4636476}{3}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $0.4636476$ by $3$ .

$x = 0.1545492$

Step 2.5

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.

$3 x = (3.14159265) + 0.4636476$

Step 2.6

Solve for $x$ .

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Step 2.6.1

Add $3.14159265$ and $0.4636476$ .

$3 x = 3.60524026$

Step 2.6.2

Divide each term in $3 x = 3.60524026$ by $3$ and simplify.

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Step 2.6.2.1

Divide each term in $3 x = 3.60524026$ by $3$ .

$\frac{3 x}{3} = \frac{3.60524026}{3}$

Step 2.6.2.2

Simplify the left side.

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Step 2.6.2.2.1

Cancel the common factor of $3$ .

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Step 2.6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{3.60524026}{3}$

Step 2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3.60524026}{3}$

Step 2.6.2.3

Simplify the right side.

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Step 2.6.2.3.1

Divide $3.60524026$ by $3$ .

$x = 1.20174675$

Step 2.7

Find the period of $\tan (3 x)$ .

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Step 2.7.1

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

$\frac{π}{| b |}$

Step 2.7.2

Replace $b$ with $3$ in the formula for period.

$\frac{π}{| 3 |}$

Step 2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{π}{3}$

Step 2.8

The period of the $\tan (3 x)$ function is $\frac{π}{3}$ so values will repeat every $\frac{π}{3}$ radians in both directions.

$x = 0.1545492 + \frac{π n}{3}, 1.20174675 + \frac{π n}{3}$ , for any integer $n$

Step 2.9

Consolidate $0.1545492 + \frac{π n}{3}$ and $1.20174675 + \frac{π n}{3}$ to $0.1545492 + \frac{π n}{3}$ .

$x = 0.1545492 + \frac{π n}{3}$ , for any integer $n$