Trigonometry Examples

$\sin (2 x) = 3 \cos (2 x)$

Step 1

Divide each term in the equation by $\cos (2 x)$ .

$\frac{\sin (2 x)}{\cos (2 x)} = \frac{3 \cos (2 x)}{\cos (2 x)}$

Step 2

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

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

Step 3

Cancel the common factor of $\cos (2 x)$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $3$ by $1$ .

$\tan (2 x) = 3$

Step 4

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

$2 x = \arctan (3)$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arctan (3)$ .

$2 x = 1.24904577$

Step 6

Divide each term in $2 x = 1.24904577$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 x = 1.24904577$ by $2$ .

$\frac{2 x}{2} = \frac{1.24904577}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1.24904577}{2}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.24904577}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $1.24904577$ by $2$ .

$x = 0.62452288$

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.

$2 x = (3.14159265) + 1.24904577$

Step 8

Solve for $x$ .

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Step 8.1

Add $3.14159265$ and $1.24904577$ .

$2 x = 4.39063842$

Step 8.2

Divide each term in $2 x = 4.39063842$ by $2$ and simplify.

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Step 8.2.1

Divide each term in $2 x = 4.39063842$ by $2$ .

$\frac{2 x}{2} = \frac{4.39063842}{2}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $2$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{4.39063842}{2}$

Step 8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4.39063842}{2}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Divide $4.39063842$ by $2$ .

$x = 2.19531921$

Step 9

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

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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 $2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 9.3

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

$\frac{π}{2}$

Step 10

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

$x = 0.62452288 + \frac{π n}{2}, 2.19531921 + \frac{π n}{2}$ , for any integer $n$

Step 11

Consolidate $0.62452288 + \frac{π n}{2}$ and $2.19531921 + \frac{π n}{2}$ to $0.62452288 + \frac{π n}{2}$ .

$x = 0.62452288 + \frac{π n}{2}$ , for any integer $n$