Calculus Examples

$- 2 \sin (t) + 2 \cos (2 t) = 0$

Step 1

Simplify each term.

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Step 1.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$- 2 \sin (t) + 2 (1 - 2 \sin^{2} (t)) = 0$

Step 1.2

Apply the distributive property.

$- 2 \sin (t) + 2 \cdot 1 + 2 (- 2 \sin^{2} (t)) = 0$

Step 1.3

Multiply $2$ by $1$ .

$- 2 \sin (t) + 2 + 2 (- 2 \sin^{2} (t)) = 0$

Step 1.4

Multiply $- 2$ by $2$ .

$- 2 \sin (t) + 2 - 4 \sin^{2} (t) = 0$

Step 2

Factor $- 2 \sin (t) + 2 - 4 \sin^{2} (t)$ .

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Step 2.1

Factor $2$ out of $- 2 \sin (t) + 2 - 4 \sin^{2} (t)$ .

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Step 2.1.1

Factor $2$ out of $- 2 \sin (t)$ .

$2 (- \sin (t)) + 2 - 4 \sin^{2} (t) = 0$

Step 2.1.2

Factor $2$ out of $2$ .

$2 (- \sin (t)) + 2 (1) - 4 \sin^{2} (t) = 0$

Step 2.1.3

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

$2 (- \sin (t)) + 2 (1) + 2 (- 2 \sin^{2} (t)) = 0$

Step 2.1.4

Factor $2$ out of $2 (- \sin (t)) + 2 (1)$ .

$2 (- \sin (t) + 1) + 2 (- 2 \sin^{2} (t)) = 0$

Step 2.1.5

Factor $2$ out of $2 (- \sin (t) + 1) + 2 (- 2 \sin^{2} (t))$ .

$2 (- \sin (t) + 1 - 2 \sin^{2} (t)) = 0$

Step 2.2

Factor.

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Step 2.2.1

Factor by grouping.

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Step 2.2.1.1

Reorder terms.

$2 (- 2 \sin^{2} (t) - \sin (t) + 1) = 0$

Step 2.2.1.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 2 \cdot 1 = - 2$ and whose sum is $b = - 1$ .

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Step 2.2.1.2.1

Factor $- 1$ out of $- \sin (t)$ .

$2 (- 2 \sin^{2} (t) - \sin (t) + 1) = 0$

Step 2.2.1.2.2

Rewrite $- 1$ as $1$ plus $- 2$

$2 (- 2 \sin^{2} (t) + (1 - 2) \sin (t) + 1) = 0$

Step 2.2.1.2.3

Apply the distributive property.

$2 (- 2 \sin^{2} (t) + 1 \sin (t) - 2 \sin (t) + 1) = 0$

Step 2.2.1.2.4

Multiply $\sin (t)$ by $1$ .

$2 (- 2 \sin^{2} (t) + \sin (t) - 2 \sin (t) + 1) = 0$

Step 2.2.1.3

Factor out the greatest common factor from each group.

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Step 2.2.1.3.1

Group the first two terms and the last two terms.

$2 (- 2 \sin^{2} (t) + \sin (t) - 2 \sin (t) + 1) = 0$

Step 2.2.1.3.2

Factor out the greatest common factor (GCF) from each group.

$2 (\sin (t) (- 2 \sin (t) + 1) + 1 (- 2 \sin (t) + 1)) = 0$

Step 2.2.1.4

Factor the polynomial by factoring out the greatest common factor, $- 2 \sin (t) + 1$ .

$2 ((- 2 \sin (t) + 1) (\sin (t) + 1)) = 0$

Step 2.2.2

Remove unnecessary parentheses.

$2 (- 2 \sin (t) + 1) (\sin (t) + 1) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$- 2 \sin (t) + 1 = 0$

$\sin (t) + 1 = 0$

Step 4

Set $- 2 \sin (t) + 1$ equal to $0$ and solve for $t$ .

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Step 4.1

Set $- 2 \sin (t) + 1$ equal to $0$ .

$- 2 \sin (t) + 1 = 0$

Step 4.2

Solve $- 2 \sin (t) + 1 = 0$ for $t$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$- 2 \sin (t) = - 1$

Step 4.2.2

Divide each term in $- 2 \sin (t) = - 1$ by $- 2$ and simplify.

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Step 4.2.2.1

Divide each term in $- 2 \sin (t) = - 1$ by $- 2$ .

$\frac{- 2 \sin (t)}{- 2} = \frac{- 1}{- 2}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 \sin (t)}{- 2} = \frac{- 1}{- 2}$

Step 4.2.2.2.1.2

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

$\sin (t) = \frac{- 1}{- 2}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Dividing two negative values results in a positive value.

$\sin (t) = \frac{1}{2}$

Step 4.2.3

Take the inverse sine of both sides of the equation to extract $t$ from inside the sine.

$t = \arcsin (\frac{1}{2})$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$t = \frac{π}{6}$

Step 4.2.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$t = π - \frac{π}{6}$

Step 4.2.6

Simplify $π - \frac{π}{6}$ .

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Step 4.2.6.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$t = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 4.2.6.2

Combine fractions.

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Step 4.2.6.2.1

Combine $π$ and $\frac{6}{6}$ .

$t = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 4.2.6.2.2

Combine the numerators over the common denominator.

$t = \frac{π \cdot 6 - π}{6}$

Step 4.2.6.3

Simplify the numerator.

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Step 4.2.6.3.1

Move $6$ to the left of $π$ .

$t = \frac{6 \cdot π - π}{6}$

Step 4.2.6.3.2

Subtract $π$ from $6 π$ .

$t = \frac{5 π}{6}$

Step 4.2.7

Find the period of $\sin (t)$ .

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Step 4.2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 4.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 4.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.8

The period of the $\sin (t)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$t = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 5

Set $\sin (t) + 1$ equal to $0$ and solve for $t$ .

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Step 5.1

Set $\sin (t) + 1$ equal to $0$ .

$\sin (t) + 1 = 0$

Step 5.2

Solve $\sin (t) + 1 = 0$ for $t$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$\sin (t) = - 1$

Step 5.2.2

Take the inverse sine of both sides of the equation to extract $t$ from inside the sine.

$t = \arcsin (- 1)$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$t = - \frac{π}{2}$

Step 5.2.4

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$t = 2 π + \frac{π}{2} + π$

Step 5.2.5

Simplify the expression to find the second solution.

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Step 5.2.5.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$t = 2 π + \frac{π}{2} + π - 2 π$

Step 5.2.5.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$t = \frac{3 π}{2}$

Step 5.2.6

Find the period of $\sin (t)$ .

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Step 5.2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.7

Add $2 π$ to every negative angle to get positive angles.

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Step 5.2.7.1

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

Step 5.2.7.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 5.2.7.3

Combine fractions.

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Step 5.2.7.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 5.2.7.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 5.2.7.4

Simplify the numerator.

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Step 5.2.7.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 5.2.7.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 5.2.7.5

List the new angles.

$t = \frac{3 π}{2}$

Step 5.2.8

The period of the $\sin (t)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$t = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 6

The final solution is all the values that make $2 (- 2 \sin (t) + 1) (\sin (t) + 1) = 0$ true.

$t = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 7

Consolidate the answers.

$t = \frac{π}{6} + \frac{2 π n}{3}$ , for any integer $n$