Trigonometry Examples

$h (t) = 100 - 4.9 t^{2}$

Step 1

Write $h (t) = 100 - 4.9 t^{2}$ as an equation.

$y = 100 - 4.9 t^{2}$

Step 2

Interchange the variables.

$t = 100 - 4.9 y^{2}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $100 - 4.9 y^{2} = t$ .

$100 - 4.9 y^{2} = t$

Step 3.2

Subtract $100$ from both sides of the equation.

$- 4.9 y^{2} = t - 100$

Step 3.3

Divide each term in $- 4.9 y^{2} = t - 100$ by $- 4.9$ and simplify.

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Step 3.3.1

Divide each term in $- 4.9 y^{2} = t - 100$ by $- 4.9$ .

$\frac{- 4.9 y^{2}}{- 4.9} = \frac{t}{- 4.9} + \frac{- 100}{- 4.9}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 4.9$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 4.9 y^{2}}{- 4.9} = \frac{t}{- 4.9} + \frac{- 100}{- 4.9}$

Step 3.3.2.1.2

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

$y^{2} = \frac{t}{- 4.9} + \frac{- 100}{- 4.9}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative in front of the fraction.

$y^{2} = - \frac{t}{4.9} + \frac{- 100}{- 4.9}$

Step 3.3.3.1.2

Multiply by $1$ .

$y^{2} = - \frac{1 t}{4.9} + \frac{- 100}{- 4.9}$

Step 3.3.3.1.3

Factor $4.9$ out of $4.9$ .

$y^{2} = - \frac{1 t}{4.9 (1)} + \frac{- 100}{- 4.9}$

Step 3.3.3.1.4

Separate fractions.

$y^{2} = - (\frac{1}{4.9} \cdot \frac{t}{1}) + \frac{- 100}{- 4.9}$

Step 3.3.3.1.5

Divide $1$ by $4.9$ .

$y^{2} = - (0.20408163 \frac{t}{1}) + \frac{- 100}{- 4.9}$

Step 3.3.3.1.6

Divide $t$ by $1$ .

$y^{2} = - (0.20408163 t) + \frac{- 100}{- 4.9}$

Step 3.3.3.1.7

Multiply $0.20408163$ by $- 1$ .

$y^{2} = - 0.20408163 t + \frac{- 100}{- 4.9}$

Step 3.3.3.1.8

Divide $- 100$ by $- 4.9$ .

$y^{2} = - 0.20408163 t + 20.40816326$

Step 3.4

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

$y = \pm \sqrt{- 0.20408163 t + 20.40816326}$

Step 3.5

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

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Step 3.5.1

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

$y = \sqrt{- 0.20408163 t + 20.40816326}$

Step 3.5.2

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

$y = - \sqrt{- 0.20408163 t + 20.40816326}$

Step 3.5.3

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

$y = \sqrt{- 0.20408163 t + 20.40816326}$

$y = - \sqrt{- 0.20408163 t + 20.40816326}$

$y = \sqrt{- 0.20408163 t + 20.40816326}$

$y = - \sqrt{- 0.20408163 t + 20.40816326}$

$y = \sqrt{- 0.20408163 t + 20.40816326}$

$y = - \sqrt{- 0.20408163 t + 20.40816326}$

Step 4

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

$h^{- 1} (t) = \sqrt{- 0.20408163 t + 20.40816326}, - \sqrt{- 0.20408163 t + 20.40816326}$

Step 5

Verify if $h^{- 1} (t) = \sqrt{- 0.20408163 t + 20.40816326}, - \sqrt{- 0.20408163 t + 20.40816326}$ is the inverse of $h (t) = 100 - 4.9 t^{2}$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $h (t) = 100 - 4.9 t^{2}$ and $h^{- 1} (t) = \sqrt{- 0.20408163 t + 20.40816326}, - \sqrt{- 0.20408163 t + 20.40816326}$ and compare them.

Step 5.2

Find the range of $h (t) = 100 - 4.9 t^{2}$ .

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Step 5.2.1

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

Interval Notation:

$(- \infty, 100]$

Step 5.3

Find the domain of $\sqrt{- 0.20408163 t + 20.40816326}$ .

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Step 5.3.1

Set the radicand in $\sqrt{- 0.20408163 t + 20.40816326}$ greater than or equal to $0$ to find where the expression is defined.

$- 0.20408163 t + 20.40816326 \geq 0$

Step 5.3.2

Solve for $t$ .

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Step 5.3.2.1

Subtract $20.40816326$ from both sides of the inequality.

$- 0.20408163 t \geq - 20.40816326$

Step 5.3.2.2

Divide each term in $- 0.20408163 t \geq - 20.40816326$ by $- 0.20408163$ and simplify.

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Step 5.3.2.2.1

Divide each term in $- 0.20408163 t \geq - 20.40816326$ by $- 0.20408163$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 0.20408163 t}{- 0.20408163} \leq \frac{- 20.40816326}{- 0.20408163}$

Step 5.3.2.2.2

Simplify the left side.

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Step 5.3.2.2.2.1

Cancel the common factor of $- 0.20408163$ .

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Step 5.3.2.2.2.1.1

Cancel the common factor.

$\frac{- 0.20408163 t}{- 0.20408163} \leq \frac{- 20.40816326}{- 0.20408163}$

Step 5.3.2.2.2.1.2

Divide $t$ by $1$ .

$t \leq \frac{- 20.40816326}{- 0.20408163}$

Step 5.3.2.2.3

Simplify the right side.

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Step 5.3.2.2.3.1

Divide $- 20.40816326$ by $- 0.20408163$ .

$t \leq 100$

Step 5.3.3

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

$(- \infty, 100]$

Step 5.4

Since the domain of $h^{- 1} (t) = \sqrt{- 0.20408163 t + 20.40816326}, - \sqrt{- 0.20408163 t + 20.40816326}$ is not equal to the range of $h (t) = 100 - 4.9 t^{2}$ , then $h^{- 1} (t) = \sqrt{- 0.20408163 t + 20.40816326}, - \sqrt{- 0.20408163 t + 20.40816326}$ is not an inverse of $h (t) = 100 - 4.9 t^{2}$ .

There is no inverse

Step 6