Trigonometry Examples

$f (x) = 3 + 35 t - 16 t^{2}$ , $f (x) = 21$

Step 1

Substitute $21$ for $f (x)$ .

$21 = 3 + 35 t - 16 t^{2}$

Step 2

Solve for $t$ .

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Step 2.1

Rewrite the equation as $3 + 35 t - 16 t^{2} = 21$ .

$3 + 35 t - 16 t^{2} = 21$

Step 2.2

Move all terms to the left side of the equation and simplify.

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Step 2.2.1

Subtract $21$ from both sides of the equation.

$3 + 35 t - 16 t^{2} - 21 = 0$

Step 2.2.2

Subtract $21$ from $3$ .

$35 t - 16 t^{2} - 18 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4

Substitute the values $a = - 16$ , $b = 35$ , and $c = - 18$ into the quadratic formula and solve for $t$ .

$\frac{- 35 \pm \sqrt{35^{2} - 4 \cdot (- 16 \cdot - 18)}}{2 \cdot - 16}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $35$ to the power of $2$ .

$t = \frac{- 35 \pm \sqrt{1225 - 4 \cdot - 16 \cdot - 18}}{2 \cdot - 16}$

Step 2.5.1.2

Multiply $- 4 \cdot - 16 \cdot - 18$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 35 \pm \sqrt{1225 + 64 \cdot - 18}}{2 \cdot - 16}$

Step 2.5.1.2.2

Multiply $64$ by $- 18$ .

$t = \frac{- 35 \pm \sqrt{1225 - 1152}}{2 \cdot - 16}$

Step 2.5.1.3

Subtract $1152$ from $1225$ .

$t = \frac{- 35 \pm \sqrt{73}}{2 \cdot - 16}$

Step 2.5.2

Multiply $2$ by $- 16$ .

$t = \frac{- 35 \pm \sqrt{73}}{- 32}$

Step 2.5.3

Simplify $\frac{- 35 \pm \sqrt{73}}{- 32}$ .

$t = \frac{35 \pm \sqrt{73}}{32}$

Step 2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $35$ to the power of $2$ .

$t = \frac{- 35 \pm \sqrt{1225 - 4 \cdot - 16 \cdot - 18}}{2 \cdot - 16}$

Step 2.6.1.2

Multiply $- 4 \cdot - 16 \cdot - 18$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 35 \pm \sqrt{1225 + 64 \cdot - 18}}{2 \cdot - 16}$

Step 2.6.1.2.2

Multiply $64$ by $- 18$ .

$t = \frac{- 35 \pm \sqrt{1225 - 1152}}{2 \cdot - 16}$

Step 2.6.1.3

Subtract $1152$ from $1225$ .

$t = \frac{- 35 \pm \sqrt{73}}{2 \cdot - 16}$

Step 2.6.2

Multiply $2$ by $- 16$ .

$t = \frac{- 35 \pm \sqrt{73}}{- 32}$

Step 2.6.3

Simplify $\frac{- 35 \pm \sqrt{73}}{- 32}$ .

$t = \frac{35 \pm \sqrt{73}}{32}$

Step 2.6.4

Change the $\pm$ to $+$ .

$t = \frac{35 + \sqrt{73}}{32}$

Step 2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $35$ to the power of $2$ .

$t = \frac{- 35 \pm \sqrt{1225 - 4 \cdot - 16 \cdot - 18}}{2 \cdot - 16}$

Step 2.7.1.2

Multiply $- 4 \cdot - 16 \cdot - 18$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 35 \pm \sqrt{1225 + 64 \cdot - 18}}{2 \cdot - 16}$

Step 2.7.1.2.2

Multiply $64$ by $- 18$ .

$t = \frac{- 35 \pm \sqrt{1225 - 1152}}{2 \cdot - 16}$

Step 2.7.1.3

Subtract $1152$ from $1225$ .

$t = \frac{- 35 \pm \sqrt{73}}{2 \cdot - 16}$

Step 2.7.2

Multiply $2$ by $- 16$ .

$t = \frac{- 35 \pm \sqrt{73}}{- 32}$

Step 2.7.3

Simplify $\frac{- 35 \pm \sqrt{73}}{- 32}$ .

$t = \frac{35 \pm \sqrt{73}}{32}$

Step 2.7.4

Change the $\pm$ to $-$ .

$t = \frac{35 - \sqrt{73}}{32}$

Step 2.8

The final answer is the combination of both solutions.

$t = \frac{35 + \sqrt{73}}{32}, \frac{35 - \sqrt{73}}{32}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$t = \frac{35 + \sqrt{73}}{32}, \frac{35 - \sqrt{73}}{32}$

Decimal Form:

$t = 1.36075011 \dots, 0.82674988 \dots$