Trigonometry Examples

$2 \sec^{2} (x)$

Step 1

Interchange the variables.

$x = 2 \sec^{2} (y)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $2 \sec^{2} (y) = x$ .

$2 \sec^{2} (y) = x$

Step 2.2

Divide each term in $2 \sec^{2} (y) = x$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 \sec^{2} (y) = x$ by $2$ .

$\frac{2 \sec^{2} (y)}{2} = \frac{x}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 \sec^{2} (y)}{2} = \frac{x}{2}$

Step 2.2.2.1.2

Divide $\sec^{2} (y)$ by $1$ .

$\sec^{2} (y) = \frac{x}{2}$

Step 2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sec (y) = \pm \sqrt{\frac{x}{2}}$

Step 2.4

Simplify $\pm \sqrt{\frac{x}{2}}$ .

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Step 2.4.1

Rewrite $\sqrt{\frac{x}{2}}$ as $\frac{\sqrt{x}}{\sqrt{2}}$ .

$\sec (y) = \pm \frac{\sqrt{x}}{\sqrt{2}}$

Step 2.4.2

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (y) = \pm \frac{\sqrt{x}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.4.3

Combine and simplify the denominator.

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Step 2.4.3.1

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.4.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.4.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.4.3.5

Add $1$ and $1$ .

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.4.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.3.6.4

Cancel the common factor of $2$ .

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Step 2.4.3.6.4.1

Cancel the common factor.

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.3.6.4.2

Rewrite the expression.

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{2^{1}}$

Step 2.4.3.6.5

Evaluate the exponent.

$\sec (y) = \pm \frac{\sqrt{x} \sqrt{2}}{2}$

Step 2.4.4

Combine using the product rule for radicals.

$\sec (y) = \pm \frac{\sqrt{x \cdot 2}}{2}$

Step 2.4.5

Reorder factors in $\pm \frac{\sqrt{x \cdot 2}}{2}$ .

$\sec (y) = \pm \frac{\sqrt{2 x}}{2}$

Step 2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$\sec (y) = \frac{\sqrt{2 x}}{2}$

Step 2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sec (y) = - \frac{\sqrt{2 x}}{2}$

Step 2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sec (y) = \frac{\sqrt{2 x}}{2}, - \frac{\sqrt{2 x}}{2}$

Step 2.6

Set up each of the solutions to solve for $y$ .

$\sec (y) = \frac{\sqrt{2 x}}{2}$

$\sec (y) = - \frac{\sqrt{2 x}}{2}$

Step 2.7

Solve for $y$ in $\sec (y) = \frac{\sqrt{2 x}}{2}$ .

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Step 2.7.1

Take the inverse secant of both sides of the equation to extract $y$ from inside the secant.

$y = arcsec (\frac{\sqrt{2 x}}{2})$

Step 2.8

Solve for $y$ in $\sec (y) = - \frac{\sqrt{2 x}}{2}$ .

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Step 2.8.1

Take the inverse secant of both sides of the equation to extract $y$ from inside the secant.

$y = arcsec (- \frac{\sqrt{2 x}}{2})$

Step 2.9

List all of the solutions.

$y = arcsec (\frac{\sqrt{2 x}}{2})$

$y = arcsec (- \frac{\sqrt{2 x}}{2})$

$y = arcsec (\frac{\sqrt{2 x}}{2})$

$y = arcsec (- \frac{\sqrt{2 x}}{2})$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = arcsec (\frac{\sqrt{2 x}}{2}), arcsec (- \frac{\sqrt{2 x}}{2})$

Step 4

Verify if $f^{- 1} (x) = arcsec (\frac{\sqrt{2 x}}{2}), arcsec (- \frac{\sqrt{2 x}}{2})$ is the inverse of $f (x) = 2 \sec^{2} (x)$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 2 \sec^{2} (x)$ and $f^{- 1} (x) = arcsec (\frac{\sqrt{2 x}}{2}), arcsec (- \frac{\sqrt{2 x}}{2})$ and compare them.

Step 4.2

Find the range of $f (x) = 2 \sec^{2} (x)$ .

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Step 4.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[2, \infty)$

Step 4.3

Find the domain of $arcsec (\frac{\sqrt{2 x}}{2})$ .

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Step 4.3.1

Set the radicand in $\sqrt{2 x}$ greater than or equal to $0$ to find where the expression is defined.

$2 x \geq 0$

Step 4.3.2

Divide each term in $2 x \geq 0$ by $2$ and simplify.

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Step 4.3.2.1

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Divide $0$ by $2$ .

$x \geq 0$

Step 4.3.3

Set the argument in $arcsec (\frac{\sqrt{2 x}}{2})$ less than or equal to $- 1$ to find where the expression is defined.

$\frac{\sqrt{2 x}}{2} \leq - 1$

Step 4.3.4

Solve for $x$ .

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Step 4.3.4.1

Multiply both sides by $2$ .

$\frac{\sqrt{2 x}}{2} \cdot 2 \leq - 1 \cdot 2$

Step 4.3.4.2

Simplify.

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Step 4.3.4.2.1

Simplify the left side.

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Step 4.3.4.2.1.1

Cancel the common factor of $2$ .

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Step 4.3.4.2.1.1.1

Cancel the common factor.

$\frac{\sqrt{2 x}}{2} \cdot 2 \leq - 1 \cdot 2$

Step 4.3.4.2.1.1.2

Rewrite the expression.

$\sqrt{2 x} \leq - 1 \cdot 2$

Step 4.3.4.2.2

Simplify the right side.

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Step 4.3.4.2.2.1

Multiply $- 1$ by $2$ .

$\sqrt{2 x} \leq - 2$

Step 4.3.4.3

Solve for $x$ .

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Step 4.3.4.3.1

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{2 x}}^{2} \leq {(- 2)}^{2}$

Step 4.3.4.3.2

Simplify each side of the inequality.

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Step 4.3.4.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

${({(2 x)}^{\frac{1}{2}})}^{2} \leq {(- 2)}^{2}$

Step 4.3.4.3.2.2

Simplify the left side.

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Step 4.3.4.3.2.2.1

Simplify ${({(2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.4.3.2.2.1.1

Multiply the exponents in ${({(2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.4.3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x)}^{\frac{1}{2} \cdot 2} \leq {(- 2)}^{2}$

Step 4.3.4.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.4.3.2.2.1.1.2.1

Cancel the common factor.

${(2 x)}^{\frac{1}{2} \cdot 2} \leq {(- 2)}^{2}$

Step 4.3.4.3.2.2.1.1.2.2

Rewrite the expression.

${(2 x)}^{1} \leq {(- 2)}^{2}$

Step 4.3.4.3.2.2.1.2

Simplify.

$2 x \leq {(- 2)}^{2}$

Step 4.3.4.3.2.3

Simplify the right side.

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Step 4.3.4.3.2.3.1

Raise $- 2$ to the power of $2$ .

$2 x \leq 4$

Step 4.3.4.3.3

Divide each term in $2 x \leq 4$ by $2$ and simplify.

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Step 4.3.4.3.3.1

Divide each term in $2 x \leq 4$ by $2$ .

$\frac{2 x}{2} \leq \frac{4}{2}$

Step 4.3.4.3.3.2

Simplify the left side.

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Step 4.3.4.3.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.4.3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{4}{2}$

Step 4.3.4.3.3.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{4}{2}$

Step 4.3.4.3.3.3

Simplify the right side.

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Step 4.3.4.3.3.3.1

Divide $4$ by $2$ .

$x \leq 2$

Step 4.3.4.4

Find the domain of $\frac{\sqrt{2 x}}{2} + 1$ .

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Step 4.3.4.4.1

Set the radicand in $\sqrt{2 x}$ greater than or equal to $0$ to find where the expression is defined.

$2 x \geq 0$

Step 4.3.4.4.2

Divide each term in $2 x \geq 0$ by $2$ and simplify.

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Step 4.3.4.4.2.1

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 4.3.4.4.2.2

Simplify the left side.

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Step 4.3.4.4.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.4.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 4.3.4.4.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Step 4.3.4.4.2.3

Simplify the right side.

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Step 4.3.4.4.2.3.1

Divide $0$ by $2$ .

$x \geq 0$

Step 4.3.4.4.3

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Step 4.3.4.5

Use each root to create test intervals.

$x < 0$

$0 < x < 2$

$x > 2$

Step 4.3.4.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 4.3.4.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 4.3.4.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 4.3.4.6.1.2

Replace $x$ with $- 2$ in the original inequality.

$\frac{\sqrt{2 (- 2)}}{2} \leq - 1$

Step 4.3.4.6.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 4.3.4.6.2

Test a value on the interval $0 < x < 2$ to see if it makes the inequality true.

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Step 4.3.4.6.2.1

Choose a value on the interval $0 < x < 2$ and see if this value makes the original inequality true.

$x = 1$

Step 4.3.4.6.2.2

Replace $x$ with $1$ in the original inequality.

$\frac{\sqrt{2 (1)}}{2} \leq - 1$

Step 4.3.4.6.2.3

The left side $0.70710678$ is greater than the right side $- 1$ , which means that the given statement is false.

False

Step 4.3.4.6.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 4.3.4.6.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 4.3.4.6.3.2

Replace $x$ with $4$ in the original inequality.

$\frac{\sqrt{2 (4)}}{2} \leq - 1$

Step 4.3.4.6.3.3

The left side $1.41421356$ is greater than the right side $- 1$ , which means that the given statement is false.

False

Step 4.3.4.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$0 < x < 2$ False

$x > 2$ False

$x < 0$ False

$0 < x < 2$ False

$x > 2$ False

Step 4.3.4.7

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 4.3.5

Set the argument in $arcsec (\frac{\sqrt{2 x}}{2})$ greater than or equal to $1$ to find where the expression is defined.

$\frac{\sqrt{2 x}}{2} \geq 1$

Step 4.3.6

Solve for $x$ .

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Step 4.3.6.1

Multiply both sides by $2$ .

$\frac{\sqrt{2 x}}{2} \cdot 2 \geq 1 \cdot 2$

Step 4.3.6.2

Simplify.

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Step 4.3.6.2.1

Simplify the left side.

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Step 4.3.6.2.1.1

Cancel the common factor of $2$ .

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Step 4.3.6.2.1.1.1

Cancel the common factor.

$\frac{\sqrt{2 x}}{2} \cdot 2 \geq 1 \cdot 2$

Step 4.3.6.2.1.1.2

Rewrite the expression.

$\sqrt{2 x} \geq 1 \cdot 2$

Step 4.3.6.2.2

Simplify the right side.

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Step 4.3.6.2.2.1

Multiply $2$ by $1$ .

$\sqrt{2 x} \geq 2$

Step 4.3.6.3

Solve for $x$ .

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Step 4.3.6.3.1

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{2 x}}^{2} \geq 2^{2}$

Step 4.3.6.3.2

Simplify each side of the inequality.

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Step 4.3.6.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

${({(2 x)}^{\frac{1}{2}})}^{2} \geq 2^{2}$

Step 4.3.6.3.2.2

Simplify the left side.

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Step 4.3.6.3.2.2.1

Simplify ${({(2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.6.3.2.2.1.1

Multiply the exponents in ${({(2 x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.6.3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x)}^{\frac{1}{2} \cdot 2} \geq 2^{2}$

Step 4.3.6.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.6.3.2.2.1.1.2.1

Cancel the common factor.

${(2 x)}^{\frac{1}{2} \cdot 2} \geq 2^{2}$

Step 4.3.6.3.2.2.1.1.2.2

Rewrite the expression.

${(2 x)}^{1} \geq 2^{2}$

Step 4.3.6.3.2.2.1.2

Simplify.

$2 x \geq 2^{2}$

Step 4.3.6.3.2.3

Simplify the right side.

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Step 4.3.6.3.2.3.1

Raise $2$ to the power of $2$ .

$2 x \geq 4$

Step 4.3.6.3.3

Divide each term in $2 x \geq 4$ by $2$ and simplify.

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Step 4.3.6.3.3.1

Divide each term in $2 x \geq 4$ by $2$ .

$\frac{2 x}{2} \geq \frac{4}{2}$

Step 4.3.6.3.3.2

Simplify the left side.

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Step 4.3.6.3.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.6.3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{4}{2}$

Step 4.3.6.3.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{4}{2}$

Step 4.3.6.3.3.3

Simplify the right side.

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Step 4.3.6.3.3.3.1

Divide $4$ by $2$ .

$x \geq 2$

Step 4.3.6.4

Find the domain of $\frac{\sqrt{2 x}}{2} - 1$ .

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Step 4.3.6.4.1

Set the radicand in $\sqrt{2 x}$ greater than or equal to $0$ to find where the expression is defined.

$2 x \geq 0$

Step 4.3.6.4.2

Divide each term in $2 x \geq 0$ by $2$ and simplify.

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Step 4.3.6.4.2.1

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 4.3.6.4.2.2

Simplify the left side.

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Step 4.3.6.4.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.6.4.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 4.3.6.4.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Step 4.3.6.4.2.3

Simplify the right side.

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Step 4.3.6.4.2.3.1

Divide $0$ by $2$ .

$x \geq 0$

Step 4.3.6.4.3

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Step 4.3.6.5

The solution consists of all of the true intervals.

$x \geq 2$

Step 4.3.7

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Step 4.4

Since the domain of $f^{- 1} (x) = arcsec (\frac{\sqrt{2 x}}{2}), arcsec (- \frac{\sqrt{2 x}}{2})$ is not equal to the range of $f (x) = 2 \sec^{2} (x)$ , then $f^{- 1} (x) = arcsec (\frac{\sqrt{2 x}}{2}), arcsec (- \frac{\sqrt{2 x}}{2})$ is not an inverse of $f (x) = 2 \sec^{2} (x)$ .

There is no inverse

Step 5