Trigonometry Examples

$3 \tan (\frac{1}{3} x) = 8$

Step 1

Divide each term in $3 \tan (\frac{1}{3} x) = 8$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 \tan (\frac{1}{3} x) = 8$ by $3$ .

$\frac{3 \tan (\frac{1}{3} x)}{3} = \frac{8}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 \tan (\frac{1}{3} x)}{3} = \frac{8}{3}$

Step 1.2.1.2

Divide $\tan (\frac{1}{3} x)$ by $1$ .

$\tan (\frac{1}{3} x) = \frac{8}{3}$

Step 2

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

$\frac{1}{3} x = \arctan (\frac{8}{3})$

Step 3

Simplify the left side.

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Step 3.1

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} = \arctan (\frac{8}{3})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arctan (\frac{8}{3})$ .

$\frac{x}{3} = 1.21202565$

Step 5

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 \cdot 1.21202565$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $3$ .

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Step 6.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 \cdot 1.21202565$

Step 6.1.1.2

Rewrite the expression.

$x = 3 \cdot 1.21202565$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $3$ by $1.21202565$ .

$x = 3.63607696$

Step 7

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.

$\frac{x}{3} = (3.14159265) + 1.21202565$

Step 8

Solve for $x$ .

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Step 8.1

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 ((3.14159265) + 1.21202565)$

Step 8.2

Simplify both sides of the equation.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Cancel the common factor of $3$ .

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Step 8.2.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 ((3.14159265) + 1.21202565)$

Step 8.2.1.1.2

Rewrite the expression.

$x = 3 ((3.14159265) + 1.21202565)$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Simplify $3 ((3.14159265) + 1.21202565)$ .

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Step 8.2.2.1.1

Add $3.14159265$ and $1.21202565$ .

$x = 3 \cdot 4.35361831$

Step 8.2.2.1.2

Multiply $3$ by $4.35361831$ .

$x = 13.06085493$

Step 9

Find the period of $\tan (\frac{1}{3} x)$ .

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Step 9.1

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

$\frac{π}{| b |}$

Step 9.2

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{π}{| \frac{1}{3} |}$

Step 9.3

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{3}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 3$

Step 9.5

Move $3$ to the left of $π$ .

$3 π$

Step 10

The period of the $\tan (\frac{1}{3} x)$ function is $3 π$ so values will repeat every $3 π$ radians in both directions.

$x = 3.63607696 + 3 π n, 13.06085493 + 3 π n$ , for any integer $n$

Step 11

Consolidate $3.63607696 + 3 π n$ and $13.06085493 + 3 π n$ to $3.63607696 + 3 π n$ .

$x = 3.63607696 + 3 π n$ , for any integer $n$