Trigonometry Examples

$- 3 \tan (2 x) + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$- 3 \tan (2 x) = - 1$

Step 2

Divide each term in $- 3 \tan (2 x) = - 1$ by $- 3$ and simplify.

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Step 2.1

Divide each term in $- 3 \tan (2 x) = - 1$ by $- 3$ .

$\frac{- 3 \tan (2 x)}{- 3} = \frac{- 1}{- 3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 3 \tan (2 x)}{- 3} = \frac{- 1}{- 3}$

Step 2.2.1.2

Divide $\tan (2 x)$ by $1$ .

$\tan (2 x) = \frac{- 1}{- 3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Dividing two negative values results in a positive value.

$\tan (2 x) = \frac{1}{3}$

Step 3

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

$2 x = \arctan (\frac{1}{3})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arctan (\frac{1}{3})$ .

$2 x = 0.32175055$

Step 5

Divide each term in $2 x = 0.32175055$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = 0.32175055$ by $2$ .

$\frac{2 x}{2} = \frac{0.32175055}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0.32175055}{2}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.32175055}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0.32175055$ by $2$ .

$x = 0.16087527$

Step 6

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.

$2 x = (3.14159265) + 0.32175055$

Step 7

Solve for $x$ .

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Step 7.1

Add $3.14159265$ and $0.32175055$ .

$2 x = 3.4633432$

Step 7.2

Divide each term in $2 x = 3.4633432$ by $2$ and simplify.

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Step 7.2.1

Divide each term in $2 x = 3.4633432$ by $2$ .

$\frac{2 x}{2} = \frac{3.4633432}{2}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3.4633432}{2}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3.4633432}{2}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $3.4633432$ by $2$ .

$x = 1.7316716$

Step 8

Find the period of $\tan (2 x)$ .

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

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

$\frac{π}{| 2 |}$

Step 8.3

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

$\frac{π}{2}$

Step 9

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

$x = 0.16087527 + \frac{π n}{2}, 1.7316716 + \frac{π n}{2}$ , for any integer $n$

Step 10

Consolidate $0.16087527 + \frac{π n}{2}$ and $1.7316716 + \frac{π n}{2}$ to $0.16087527 + \frac{π n}{2}$ .

$x = 0.16087527 + \frac{π n}{2}$ , for any integer $n$