Precalculus Examples

$\frac{\tan (x)}{\sec (- x)}$

Step 1

Write $\frac{\tan (x)}{\sec (- x)}$ as an equation.

$y = \frac{\tan (x)}{\sec (- x)}$

Step 2

The parent function is the simplest form of the type of function given.

$y = \tan (x)$

Step 3

Simplify $\frac{\tan (x)}{\sec (- x)}$ .

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Step 3.1

Rewrite $\sec (- x)$ in terms of sines and cosines.

$y = \frac{\tan (x)}{\frac{1}{\cos (- x)}}$

Step 3.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 3.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (- x)}$ .

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

Step 3.4

Write $\cos (- x)$ as a fraction with denominator $1$ .

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

Step 3.5

Simplify.

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Step 3.5.1

Divide $\cos (- x)$ by $1$ .

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

Step 3.5.2

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

$y = \tan (x) \cos (- x)$

Step 3.6

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

$y = \tan (x) \cos (x)$

Step 3.7

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.7.1

Rewrite $\tan (x) \cos (x)$ in terms of sines and cosines.

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

Step 3.7.2

Cancel the common factors.

$y = \sin (x)$

Step 4

Assume that $y = \tan (x)$ is $f (x) = \tan (x)$ and $y = \frac{\tan (x)}{\sec (- x)}$ is $g (x) = \sin (x)$ .

$f (x) = \tan (x)$

$g (x) = \sin (x)$

Step 5

Use the form $a \sin (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 1$

$b = 1$

$c = 0$

$d = 0$

Step 6

Find the amplitude $| a |$ .

Amplitude: $1$

Step 7

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

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Step 7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 7.3

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

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 8.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 8.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{1}$

Step 8.3

Divide $0$ by $1$ .

Phase Shift: $0$

Step 9

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2 π$

Phase Shift: None

Vertical Shift: None

Step 10