Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$\tan (x) = 1$

Step 4

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

$x = \arctan (1)$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arctan (1)$ is $45$ .

$x = 45$

Step 6

The tangent function is positive in the first and third quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the fourth quadrant.

$x = 180 + 45$

Step 7

Add $180$ and $45$ .

$x = 225$

Step 8

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

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Step 8.1

The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 8.2

Replace $b$ with $1$ in the formula for period.

$\frac{180}{| 1 |}$

Step 8.3

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

$\frac{180}{1}$

Step 8.4

Divide $180$ by $1$ .

$180$

Step 9

The period of the $\tan (x)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$x = 45 + 180 n, 225 + 180 n$ , for any integer $n$

Step 10

Consolidate the answers.

$x = 45 + 180 n$ , for any integer $n$