Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $3$ by $1$ .

$\tan (x) = 3$

Step 4

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

$x = \arctan (3)$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arctan (3)$ .

$x = 1.24904577$

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.

$x = (3.14159265) + 1.24904577$

Step 7

Solve for $x$ .

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Step 7.1

Remove parentheses.

$x = 3.14159265 + 1.24904577$

Step 7.2

Remove parentheses.

$x = (3.14159265) + 1.24904577$

Step 7.3

Add $3.14159265$ and $1.24904577$ .

$x = 4.39063842$

Step 8

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

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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 $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 8.4

Divide $π$ by $1$ .

$π$

Step 9

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

$x = 1.24904577 + π n, 4.39063842 + π n$ , for any integer $n$

Step 10

Consolidate $1.24904577 + π n$ and $4.39063842 + π n$ to $1.24904577 + π n$ .

$x = 1.24904577 + π n$ , for any integer $n$