Trigonometry Examples

$20 = \frac{1}{2} \cdot ((- 32) t^{2}) + 60 t + 60$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (- 32 t^{2}) + 60 t + 60 = 20$ .

$\frac{1}{2} \cdot (- 32 t^{2}) + 60 t + 60 = 20$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Factor $2$ out of $- 32 t^{2}$ .

$\frac{1}{2} \cdot (2 (- 16 t^{2})) + 60 t + 60 = 20$

Step 2.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 (- 16 t^{2})) + 60 t + 60 = 20$

Step 2.3

Rewrite the expression.

$- 16 t^{2} + 60 t + 60 = 20$

Step 3

Subtract $20$ from both sides of the equation.

$- 16 t^{2} + 60 t + 60 - 20 = 0$

Step 4

Subtract $20$ from $60$ .

$- 16 t^{2} + 60 t + 40 = 0$

Step 5

Factor $- 4$ out of $- 16 t^{2} + 60 t + 40$ .

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Step 5.1

Factor $- 4$ out of $- 16 t^{2}$ .

$- 4 (4 t^{2}) + 60 t + 40 = 0$

Step 5.2

Factor $- 4$ out of $60 t$ .

$- 4 (4 t^{2}) - 4 (- 15 t) + 40 = 0$

Step 5.3

Factor $- 4$ out of $40$ .

$- 4 (4 t^{2}) - 4 (- 15 t) - 4 \cdot - 10 = 0$

Step 5.4

Factor $- 4$ out of $- 4 (4 t^{2}) - 4 (- 15 t)$ .

$- 4 (4 t^{2} - 15 t) - 4 \cdot - 10 = 0$

Step 5.5

Factor $- 4$ out of $- 4 (4 t^{2} - 15 t) - 4 (- 10)$ .

$- 4 (4 t^{2} - 15 t - 10) = 0$

Step 6

Divide each term in $- 4 (4 t^{2} - 15 t - 10) = 0$ by $- 4$ and simplify.

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Step 6.1

Divide each term in $- 4 (4 t^{2} - 15 t - 10) = 0$ by $- 4$ .

$\frac{- 4 (4 t^{2} - 15 t - 10)}{- 4} = \frac{0}{- 4}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 4$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 4 (4 t^{2} - 15 t - 10)}{- 4} = \frac{0}{- 4}$

Step 6.2.1.2

Divide $4 t^{2} - 15 t - 10$ by $1$ .

$4 t^{2} - 15 t - 10 = \frac{0}{- 4}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $- 4$ .

$4 t^{2} - 15 t - 10 = 0$

Step 7

Use the quadratic formula to find the solutions.

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

Step 8

Substitute the values $a = 4$ , $b = - 15$ , and $c = - 10$ into the quadratic formula and solve for $t$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (4 \cdot - 10)}}{2 \cdot 4}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 15$ to the power of $2$ .

$t = \frac{15 \pm \sqrt{225 - 4 \cdot 4 \cdot - 10}}{2 \cdot 4}$

Step 9.1.2

Multiply $- 4 \cdot 4 \cdot - 10$ .

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Step 9.1.2.1

Multiply $- 4$ by $4$ .

$t = \frac{15 \pm \sqrt{225 - 16 \cdot - 10}}{2 \cdot 4}$

Step 9.1.2.2

Multiply $- 16$ by $- 10$ .

$t = \frac{15 \pm \sqrt{225 + 160}}{2 \cdot 4}$

Step 9.1.3

Add $225$ and $160$ .

$t = \frac{15 \pm \sqrt{385}}{2 \cdot 4}$

Step 9.2

Multiply $2$ by $4$ .

$t = \frac{15 \pm \sqrt{385}}{8}$

Step 10

The final answer is the combination of both solutions.

$t = \frac{15 + \sqrt{385}}{8}, \frac{15 - \sqrt{385}}{8}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$t = \frac{15 + \sqrt{385}}{8}, \frac{15 - \sqrt{385}}{8}$

Decimal Form:

$t = 4.32767710 \dots, - 0.57767710 \dots$