Trigonometry Examples

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

Step 1

Simplify the right side.

Tap for more steps...

Step 1.1

Simplify $- \sin (- x)$ .

Tap for more steps...

Step 1.1.1

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

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

Step 1.1.2

Multiply $- - \sin (x)$ .

Tap for more steps...

Step 1.1.2.1

Multiply $- 1$ by $- 1$ .

$\cos (x) = 1 \sin (x)$

Step 1.1.2.2

Multiply $\sin (x)$ by $1$ .

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

Step 2

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

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

Step 3

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

Tap for more steps...

Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$1 = \frac{\sin (x)}{\cos (x)}$

Step 4

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

$1 = \tan (x)$

Step 5

Rewrite the equation as $\tan (x) = 1$ .

$\tan (x) = 1$

Step 6

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

$x = \arctan (1)$

Step 7

Simplify the right side.

Tap for more steps...

Step 7.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 8

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.

$x = π + \frac{π}{4}$

Step 9

Simplify $π + \frac{π}{4}$ .

Tap for more steps...

Step 9.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 9.2

Combine fractions.

Tap for more steps...

Step 9.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 9.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 9.3

Simplify the numerator.

Tap for more steps...

Step 9.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 9.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 10

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

Tap for more steps...

Step 10.1

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

$\frac{π}{| b |}$

Step 10.2

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

$\frac{π}{| 1 |}$

Step 10.3

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

$\frac{π}{1}$

Step 10.4

Divide $π$ by $1$ .

$π$

Step 11

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

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 12

Consolidate the answers.

$x = \frac{π}{4} + π n$ , for any integer $n$