Trigonometry Examples

$y = \arcsin (5 - 3 x^{2})$

Step 1

Interchange the variables.

$x = \arcsin (5 - 3 y^{2})$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\arcsin (5 - 3 y^{2}) = x$ .

$\arcsin (5 - 3 y^{2}) = x$

Step 2.2

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

$5 - 3 y^{2} = \sin (x)$

Step 2.3

Subtract $5$ from both sides of the equation.

$- 3 y^{2} = \sin (x) - 5$

Step 2.4

Divide each term in $- 3 y^{2} = \sin (x) - 5$ by $- 3$ and simplify.

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Step 2.4.1

Divide each term in $- 3 y^{2} = \sin (x) - 5$ by $- 3$ .

$\frac{- 3 y^{2}}{- 3} = \frac{\sin (x)}{- 3} + \frac{- 5}{- 3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $- 3$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{- 3 y^{2}}{- 3} = \frac{\sin (x)}{- 3} + \frac{- 5}{- 3}$

Step 2.4.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{\sin (x)}{- 3} + \frac{- 5}{- 3}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Move the negative in front of the fraction.

$y^{2} = - \frac{\sin (x)}{3} + \frac{- 5}{- 3}$

Step 2.4.3.1.2

Dividing two negative values results in a positive value.

$y^{2} = - \frac{\sin (x)}{3} + \frac{5}{3}$

Step 2.5

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

$y = \pm \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

Step 2.6

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

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Step 2.6.1

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

$y = \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

Step 2.6.2

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

$y = - \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

Step 2.6.3

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

$y = \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

$y = - \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

$y = \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

$y = - \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

$y = \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

$y = - \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

Step 3

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

$f^{- 1} (x) = \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}, - \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$

Step 4

Verify if $f^{- 1} (x) = \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}, - \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$ is the inverse of $f (x) = \arcsin (5 - 3 x^{2})$ .

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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 $f (x) = \arcsin (5 - 3 x^{2})$ and $f^{- 1} (x) = \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}, - \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$ and compare them.

Step 4.2

Find the range of $f (x) = \arcsin (5 - 3 x^{2})$ .

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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:

$[- \frac{π}{2}, \frac{π}{2}]$

Step 4.3

Find the domain of $\sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$ .

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Step 4.3.1

Set the radicand in $\sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$ greater than or equal to $0$ to find where the expression is defined.

$- \frac{\sin (x)}{3} + \frac{5}{3} \geq 0$

Step 4.3.2

Solve for $x$ .

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Step 4.3.2.1

Subtract $\frac{5}{3}$ from both sides of the inequality.

$- \frac{\sin (x)}{3} \geq - \frac{5}{3}$

Step 4.3.2.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- \sin (x) \geq - 5$

Step 4.3.2.3

Divide each term in $- \sin (x) \geq - 5$ by $- 1$ and simplify.

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Step 4.3.2.3.1

Divide each term in $- \sin (x) \geq - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \sin (x)}{- 1} \leq \frac{- 5}{- 1}$

Step 4.3.2.3.2

Simplify the left side.

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Step 4.3.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (x)}{1} \leq \frac{- 5}{- 1}$

Step 4.3.2.3.2.2

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

$\sin (x) \leq \frac{- 5}{- 1}$

Step 4.3.2.3.3

Simplify the right side.

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Step 4.3.2.3.3.1

Divide $- 5$ by $- 1$ .

$\sin (x) \leq 5$

Step 4.3.2.4

The range of sine is $- 1 \leq y \leq 1$ . Since $5$ does not fall in this range, there is no solution.

No solution

Step 4.3.3

The domain is all real numbers.

$(- \infty, \infty)$

Step 4.4

Since the domain of $f^{- 1} (x) = \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}, - \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$ is not equal to the range of $f (x) = \arcsin (5 - 3 x^{2})$ , then $f^{- 1} (x) = \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}, - \sqrt{- \frac{\sin (x)}{3} + \frac{5}{3}}$ is not an inverse of $f (x) = \arcsin (5 - 3 x^{2})$ .

There is no inverse

Step 5