Trigonometry Examples

$\tan (2 x + \frac{π}{6}) = 7.57$

Step 1

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

$2 x + \frac{π}{6} = \arctan (7.57)$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arctan (7.57)$ .

$2 x + \frac{π}{6} = 1.43945639$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$2 x = 1.43945639 - \frac{π}{6}$

Step 3.2

Subtract $\frac{π}{6}$ from $1.43945639$ .

$2 x = 0.91585761$

Step 4

Divide each term in $2 x = 0.91585761$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = 0.91585761$ by $2$ .

$\frac{2 x}{2} = \frac{0.91585761}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0.91585761}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.91585761}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0.91585761$ by $2$ .

$x = 0.4579288$

Step 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.

$2 x + \frac{π}{6} = (3.14159265) + 1.43945639$

Step 6

Solve for $x$ .

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Step 6.1

Add $3.14159265$ and $1.43945639$ .

$2 x + \frac{π}{6} = 4.58104904$

Step 6.2

Move all terms not containing $x$ to the right side of the equation.

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Step 6.2.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$2 x = 4.58104904 - \frac{π}{6}$

Step 6.2.2

Subtract $\frac{π}{6}$ from $4.58104904$ .

$2 x = 4.05745026$

Step 6.3

Divide each term in $2 x = 4.05745026$ by $2$ and simplify.

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Step 6.3.1

Divide each term in $2 x = 4.05745026$ by $2$ .

$\frac{2 x}{2} = \frac{4.05745026}{2}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $2$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4.05745026}{2}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4.05745026}{2}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $4.05745026$ by $2$ .

$x = 2.02872513$

Step 7

Find the period of $\tan (2 x + \frac{π}{6})$ .

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Step 7.1

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

$\frac{π}{| b |}$

Step 7.2

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 8

The period of the $\tan (2 x + \frac{π}{6})$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = 0.4579288 + \frac{π n}{2}, 2.02872513 + \frac{π n}{2}$ , for any integer $n$

Step 9

Consolidate $0.4579288 + \frac{π n}{2}$ and $2.02872513 + \frac{π n}{2}$ to $0.4579288 + \frac{π n}{2}$ .

$x = 0.4579288 + \frac{π n}{2}$ , for any integer $n$