Precalculus Examples

$\sin^{2} (3 x) - \sin^{2} (x) = 0$

Step 1

Simplify the left side of the equation.

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Step 1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (3 x)$ and $b = \sin (x)$ .

$(\sin (3 x) + \sin (x)) (\sin (3 x) - \sin (x)) = 0$

Step 1.2

Apply the sine triple-angle identity.

$(- 4 \sin^{3} (x) + 3 \sin (x) + \sin (x)) (\sin (3 x) - \sin (x)) = 0$

Step 1.3

Add $3 \sin (x)$ and $\sin (x)$ .

$(- 4 \sin^{3} (x) + 4 \sin (x)) (\sin (3 x) - \sin (x)) = 0$

Step 1.4

Apply the sine triple-angle identity.

$(- 4 \sin^{3} (x) + 4 \sin (x)) (- 4 \sin^{3} (x) + 3 \sin (x) - \sin (x)) = 0$

Step 1.5

Subtract $\sin (x)$ from $3 \sin (x)$ .

$(- 4 \sin^{3} (x) + 4 \sin (x)) (- 4 \sin^{3} (x) + 2 \sin (x)) = 0$

Step 1.6

Expand $(- 4 \sin^{3} (x) + 4 \sin (x)) (- 4 \sin^{3} (x) + 2 \sin (x))$ using the FOIL Method.

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Step 1.6.1

Apply the distributive property.

$- 4 \sin^{3} (x) (- 4 \sin^{3} (x) + 2 \sin (x)) + 4 \sin (x) (- 4 \sin^{3} (x) + 2 \sin (x)) = 0$

Step 1.6.2

Apply the distributive property.

$- 4 \sin^{3} (x) (- 4 \sin^{3} (x)) - 4 \sin^{3} (x) (2 \sin (x)) + 4 \sin (x) (- 4 \sin^{3} (x) + 2 \sin (x)) = 0$

Step 1.6.3

Apply the distributive property.

$- 4 \sin^{3} (x) (- 4 \sin^{3} (x)) - 4 \sin^{3} (x) (2 \sin (x)) + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7

Simplify and combine like terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Rewrite using the commutative property of multiplication.

$- 4 \cdot (- 4 \sin^{3} (x) \sin^{3} (x)) - 4 \sin^{3} (x) (2 \sin (x)) + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.2

Multiply $\sin^{3} (x)$ by $\sin^{3} (x)$ by adding the exponents.

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Step 1.7.1.2.1

Move $\sin^{3} (x)$ .

$- 4 \cdot (- 4 (\sin^{3} (x) \sin^{3} (x))) - 4 \sin^{3} (x) (2 \sin (x)) + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 4 \cdot (- 4 {\sin (x)}^{3 + 3}) - 4 \sin^{3} (x) (2 \sin (x)) + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.2.3

Add $3$ and $3$ .

$- 4 \cdot (- 4 \sin^{6} (x)) - 4 \sin^{3} (x) (2 \sin (x)) + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.3

Multiply $- 4$ by $- 4$ .

$16 \sin^{6} (x) - 4 \sin^{3} (x) (2 \sin (x)) + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.4

Multiply $\sin^{3} (x)$ by $\sin (x)$ by adding the exponents.

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Step 1.7.1.4.1

Move $\sin (x)$ .

$16 \sin^{6} (x) - 4 (\sin (x) \sin^{3} (x)) \cdot 2 + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.4.2

Multiply $\sin (x)$ by $\sin^{3} (x)$ .

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Step 1.7.1.4.2.1

Raise $\sin (x)$ to the power of $1$ .

$16 \sin^{6} (x) - 4 (\sin (x) \sin^{3} (x)) \cdot 2 + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 \sin^{6} (x) - 4 {\sin (x)}^{1 + 3} \cdot 2 + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.4.3

Add $1$ and $3$ .

$16 \sin^{6} (x) - 4 \sin^{4} (x) \cdot 2 + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.5

Multiply $2$ by $- 4$ .

$16 \sin^{6} (x) - 8 \sin^{4} (x) + 4 \sin (x) (- 4 \sin^{3} (x)) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.6

Multiply $\sin (x)$ by $\sin^{3} (x)$ by adding the exponents.

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Step 1.7.1.6.1

Move $\sin^{3} (x)$ .

$16 \sin^{6} (x) - 8 \sin^{4} (x) + 4 (\sin^{3} (x) \sin (x)) \cdot - 4 + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.6.2

Multiply $\sin^{3} (x)$ by $\sin (x)$ .

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Step 1.7.1.6.2.1

Raise $\sin (x)$ to the power of $1$ .

$16 \sin^{6} (x) - 8 \sin^{4} (x) + 4 (\sin^{3} (x) \sin (x)) \cdot - 4 + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 \sin^{6} (x) - 8 \sin^{4} (x) + 4 {\sin (x)}^{3 + 1} \cdot - 4 + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.6.3

Add $3$ and $1$ .

$16 \sin^{6} (x) - 8 \sin^{4} (x) + 4 \sin^{4} (x) \cdot - 4 + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.7

Multiply $- 4$ by $4$ .

$16 \sin^{6} (x) - 8 \sin^{4} (x) - 16 \sin^{4} (x) + 4 \sin (x) (2 \sin (x)) = 0$

Step 1.7.1.8

Multiply $4 \sin (x) (2 \sin (x))$ .

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Step 1.7.1.8.1

Multiply $2$ by $4$ .

$16 \sin^{6} (x) - 8 \sin^{4} (x) - 16 \sin^{4} (x) + 8 \sin (x) \sin (x) = 0$

Step 1.7.1.8.2

Raise $\sin (x)$ to the power of $1$ .

$16 \sin^{6} (x) - 8 \sin^{4} (x) - 16 \sin^{4} (x) + 8 (\sin (x) \sin (x)) = 0$

Step 1.7.1.8.3

Raise $\sin (x)$ to the power of $1$ .

$16 \sin^{6} (x) - 8 \sin^{4} (x) - 16 \sin^{4} (x) + 8 (\sin (x) \sin (x)) = 0$

Step 1.7.1.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 \sin^{6} (x) - 8 \sin^{4} (x) - 16 \sin^{4} (x) + 8 {\sin (x)}^{1 + 1} = 0$

Step 1.7.1.8.5

Add $1$ and $1$ .

$16 \sin^{6} (x) - 8 \sin^{4} (x) - 16 \sin^{4} (x) + 8 \sin^{2} (x) = 0$

Step 1.7.2

Subtract $16 \sin^{4} (x)$ from $- 8 \sin^{4} (x)$ .

$16 \sin^{6} (x) - 24 \sin^{4} (x) + 8 \sin^{2} (x) = 0$

Step 2

Factor $16 \sin^{6} (x) - 24 \sin^{4} (x) + 8 \sin^{2} (x)$ .

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Step 2.1

Factor $8 \sin^{2} (x)$ out of $16 \sin^{6} (x) - 24 \sin^{4} (x) + 8 \sin^{2} (x)$ .

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Step 2.1.1

Factor $8 \sin^{2} (x)$ out of $16 \sin^{6} (x)$ .

$8 \sin^{2} (x) (2 \sin^{4} (x)) - 24 \sin^{4} (x) + 8 \sin^{2} (x) = 0$

Step 2.1.2

Factor $8 \sin^{2} (x)$ out of $- 24 \sin^{4} (x)$ .

$8 \sin^{2} (x) (2 \sin^{4} (x)) + 8 \sin^{2} (x) (- 3 \sin^{2} (x)) + 8 \sin^{2} (x) = 0$

Step 2.1.3

Factor $8 \sin^{2} (x)$ out of $8 \sin^{2} (x)$ .

$8 \sin^{2} (x) (2 \sin^{4} (x)) + 8 \sin^{2} (x) (- 3 \sin^{2} (x)) + 8 \sin^{2} (x) \cdot 1 = 0$

Step 2.1.4

Factor $8 \sin^{2} (x)$ out of $8 \sin^{2} (x) (2 \sin^{4} (x)) + 8 \sin^{2} (x) (- 3 \sin^{2} (x))$ .

$8 \sin^{2} (x) (2 \sin^{4} (x) - 3 \sin^{2} (x)) + 8 \sin^{2} (x) \cdot 1 = 0$

Step 2.1.5

Factor $8 \sin^{2} (x)$ out of $8 \sin^{2} (x) (2 \sin^{4} (x) - 3 \sin^{2} (x)) + 8 \sin^{2} (x) \cdot 1$ .

$8 \sin^{2} (x) (2 \sin^{4} (x) - 3 \sin^{2} (x) + 1) = 0$

Step 2.2

Rewrite $\sin^{4} (x)$ as ${(\sin^{2} (x))}^{2}$ .

$8 \sin^{2} (x) (2 {(\sin^{2} (x))}^{2} - 3 \sin^{2} (x) + 1) = 0$

Step 2.3

Let $u = \sin^{2} (x)$ . Substitute $u$ for all occurrences of $\sin^{2} (x)$ .

$8 \sin^{2} (x) (2 u^{2} - 3 u + 1) = 0$

Step 2.4

Factor by grouping.

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Step 2.4.1

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 $b = - 3$ .

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Step 2.4.1.1

Factor $- 3$ out of $- 3 u$ .

$8 \sin^{2} (x) (2 u^{2} - 3 u + 1) = 0$

Step 2.4.1.2

Rewrite $- 3$ as $- 1$ plus $- 2$

$8 \sin^{2} (x) (2 u^{2} + (- 1 - 2) u + 1) = 0$

Step 2.4.1.3

Apply the distributive property.

$8 \sin^{2} (x) (2 u^{2} - 1 u - 2 u + 1) = 0$

Step 2.4.2

Factor out the greatest common factor from each group.

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Step 2.4.2.1

Group the first two terms and the last two terms.

$8 \sin^{2} (x) ((2 u^{2} - 1 u) - 2 u + 1) = 0$

Step 2.4.2.2

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

$8 \sin^{2} (x) (u (2 u - 1) - (2 u - 1)) = 0$

Step 2.4.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$8 \sin^{2} (x) ((2 u - 1) (u - 1)) = 0$

Step 2.5

Replace all occurrences of $u$ with $\sin^{2} (x)$ .

$8 \sin^{2} (x) ((2 \sin^{2} (x) - 1) (\sin^{2} (x) - 1)) = 0$

Step 2.6

Rewrite $1$ as $1^{2}$ .

$8 \sin^{2} (x) ((2 \sin^{2} (x) - 1) (\sin^{2} (x) - 1^{2})) = 0$

Step 2.7

Factor.

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Step 2.7.1

Factor.

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Step 2.7.1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = 1$ .

$8 \sin^{2} (x) ((2 \sin^{2} (x) - 1) ((\sin (x) + 1) (\sin (x) - 1))) = 0$

Step 2.7.1.2

Remove unnecessary parentheses.

$8 \sin^{2} (x) ((2 \sin^{2} (x) - 1) (\sin (x) + 1) (\sin (x) - 1)) = 0$

Step 2.7.2

Remove unnecessary parentheses.

$8 \sin^{2} (x) (2 \sin^{2} (x) - 1) (\sin (x) + 1) (\sin (x) - 1) = 0$

Step 3

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

$\sin^{2} (x) = 0$

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

$\sin (x) + 1 = 0$

$\sin (x) - 1 = 0$

Step 4

Set $\sin^{2} (x)$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\sin^{2} (x)$ equal to $0$ .

$\sin^{2} (x) = 0$

Step 4.2

Solve $\sin^{2} (x) = 0$ for $x$ .

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Step 4.2.1

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

$\sin (x) = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.2.1

Rewrite $0$ as $0^{2}$ .

$\sin (x) = \pm \sqrt{0^{2}}$

Step 4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\sin (x) = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$\sin (x) = 0$

Step 4.2.3

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

$x = \arcsin (0)$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

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.

$x = π - 0$

Step 4.2.6

Subtract $0$ from $π$ .

$x = π$

Step 4.2.7

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

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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 $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 5

Set $2 \sin^{2} (x) - 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $2 \sin^{2} (x) - 1$ equal to $0$ .

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

Step 5.2

Solve $2 \sin^{2} (x) - 1 = 0$ for $x$ .

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Step 5.2.1

Add $1$ to both sides of the equation.

$2 \sin^{2} (x) = 1$

Step 5.2.2

Divide each term in $2 \sin^{2} (x) = 1$ by $2$ and simplify.

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Step 5.2.2.1

Divide each term in $2 \sin^{2} (x) = 1$ by $2$ .

$\frac{2 \sin^{2} (x)}{2} = \frac{1}{2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{2 \sin^{2} (x)}{2} = \frac{1}{2}$

Step 5.2.2.2.1.2

Divide $\sin^{2} (x)$ by $1$ .

$\sin^{2} (x) = \frac{1}{2}$

Step 5.2.3

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

$\sin (x) = \pm \sqrt{\frac{1}{2}}$

Step 5.2.4

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 5.2.4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\sin (x) = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 5.2.4.2

Any root of $1$ is $1$ .

$\sin (x) = \pm \frac{1}{\sqrt{2}}$

Step 5.2.4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x) = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.2.4.4

Combine and simplify the denominator.

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Step 5.2.4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x) = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.2.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 5.2.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\sin (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 5.2.4.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.2.4.4.5

Add $1$ and $1$ .

$\sin (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.2.4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sin (x) = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.2.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (x) = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.2.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sin (x) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 5.2.4.4.6.4.1

Cancel the common factor.

$\sin (x) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.2.4.4.6.4.2

Rewrite the expression.

$\sin (x) = \pm \frac{\sqrt{2}}{2^{1}}$

Step 5.2.4.4.6.5

Evaluate the exponent.

$\sin (x) = \pm \frac{\sqrt{2}}{2}$

Step 5.2.5

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

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Step 5.2.5.1

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

$\sin (x) = \frac{\sqrt{2}}{2}$

Step 5.2.5.2

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

$\sin (x) = - \frac{\sqrt{2}}{2}$

Step 5.2.5.3

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

$\sin (x) = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 5.2.6

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{\sqrt{2}}{2}$

$\sin (x) = - \frac{\sqrt{2}}{2}$

Step 5.2.7

Solve for $x$ in $\sin (x) = \frac{\sqrt{2}}{2}$ .

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Step 5.2.7.1

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

$x = \arcsin (\frac{\sqrt{2}}{2})$

Step 5.2.7.2

Simplify the right side.

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Step 5.2.7.2.1

The exact value of $\arcsin (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 5.2.7.3

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.

$x = π - \frac{π}{4}$

Step 5.2.7.4

Simplify $π - \frac{π}{4}$ .

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Step 5.2.7.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.2.7.4.2

Combine fractions.

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Step 5.2.7.4.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} - \frac{π}{4}$

Step 5.2.7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 - π}{4}$

Step 5.2.7.4.3

Simplify the numerator.

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Step 5.2.7.4.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π - π}{4}$

Step 5.2.7.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{4}$

Step 5.2.7.5

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

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Step 5.2.7.5.1

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

$\frac{2 π}{| b |}$

Step 5.2.7.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.7.5.3

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

$\frac{2 π}{1}$

Step 5.2.7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.7.6

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

$x = \frac{π}{4} + 2 π n, \frac{3 π}{4} + 2 π n$ , for any integer $n$

Step 5.2.8

Solve for $x$ in $\sin (x) = - \frac{\sqrt{2}}{2}$ .

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Step 5.2.8.1

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

$x = \arcsin (- \frac{\sqrt{2}}{2})$

Step 5.2.8.2

Simplify the right side.

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Step 5.2.8.2.1

The exact value of $\arcsin (- \frac{\sqrt{2}}{2})$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 5.2.8.3

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.

$x = 2 π + \frac{π}{4} + π$

Step 5.2.8.4

Simplify the expression to find the second solution.

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Step 5.2.8.4.1

Subtract $2 π$ from $2 π + \frac{π}{4} + π$ .

$x = 2 π + \frac{π}{4} + π - 2 π$

Step 5.2.8.4.2

The resulting angle of $\frac{5 π}{4}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{4} + π$ .

$x = \frac{5 π}{4}$

Step 5.2.8.5

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

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Step 5.2.8.5.1

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

$\frac{2 π}{| b |}$

Step 5.2.8.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.8.5.3

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

$\frac{2 π}{1}$

Step 5.2.8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.8.6

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

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Step 5.2.8.6.1

Add $2 π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + 2 π$

Step 5.2.8.6.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.2.8.6.3

Combine fractions.

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Step 5.2.8.6.3.1

Combine $2 π$ and $\frac{4}{4}$ .

$\frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 5.2.8.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 4 - π}{4}$

Step 5.2.8.6.4

Simplify the numerator.

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Step 5.2.8.6.4.1

Multiply $4$ by $2$ .

$\frac{8 π - π}{4}$

Step 5.2.8.6.4.2

Subtract $π$ from $8 π$ .

$\frac{7 π}{4}$

Step 5.2.8.6.5

List the new angles.

$x = \frac{7 π}{4}$

Step 5.2.8.7

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

$x = \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 5.2.9

List all of the solutions.

$x = \frac{π}{4} + 2 π n, \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 5.2.10

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 6

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

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Step 6.1

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

$\sin (x) + 1 = 0$

Step 6.2

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

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$\sin (x) = - 1$

Step 6.2.2

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

$x = \arcsin (- 1)$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

$x = - \frac{π}{2}$

Step 6.2.4

$x = 2 π + \frac{π}{2} + π$

Step 6.2.5

Simplify the expression to find the second solution.

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Step 6.2.5.1

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

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 6.2.5.2

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

$x = \frac{3 π}{2}$

Step 6.2.6

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

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Step 6.2.6.1

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

$\frac{2 π}{| b |}$

Step 6.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 6.2.6.3

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

$\frac{2 π}{1}$

Step 6.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.2.7

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

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Step 6.2.7.1

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

$- \frac{π}{2} + 2 π$

Step 6.2.7.2

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 6.2.7.3

Combine fractions.

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Step 6.2.7.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 6.2.7.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 6.2.7.4

Simplify the numerator.

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Step 6.2.7.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 6.2.7.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 6.2.7.5

List the new angles.

$x = \frac{3 π}{2}$

Step 6.2.8

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

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

Step 7

Set $\sin (x) - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $\sin (x) - 1$ equal to $0$ .

$\sin (x) - 1 = 0$

Step 7.2

Solve $\sin (x) - 1 = 0$ for $x$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$\sin (x) = 1$

Step 7.2.2

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

$x = \arcsin (1)$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

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

$x = \frac{π}{2}$

Step 7.2.4

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.

$x = π - \frac{π}{2}$

Step 7.2.5

Simplify $π - \frac{π}{2}$ .

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Step 7.2.5.1

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

$x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 7.2.5.2

Combine fractions.

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Step 7.2.5.2.1

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 7.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 7.2.5.3

Simplify the numerator.

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Step 7.2.5.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 7.2.5.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 7.2.6

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

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Step 7.2.6.1

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

$\frac{2 π}{| b |}$

Step 7.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 7.2.6.3

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

$\frac{2 π}{1}$

Step 7.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.2.7

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

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

Step 8

The final solution is all the values that make $8 \sin^{2} (x) (2 \sin^{2} (x) - 1) (\sin (x) + 1) (\sin (x) - 1) = 0$ true.

$x = 2 π n, π + 2 π n, \frac{π}{4} + \frac{π n}{2}, \frac{3 π}{2} + 2 π n, \frac{π}{2} + 2 π n$ , for any integer $n$

Step 9

Consolidate the answers.

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Step 9.1

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

$x = π n, \frac{π}{4} + \frac{π n}{2}, \frac{3 π}{2} + 2 π n, \frac{π}{2} + 2 π n$ , for any integer $n$

Step 9.2

Consolidate $\frac{3 π}{2} + 2 π n$ and $\frac{π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

$x = π n, \frac{π}{4} + \frac{π n}{2}, \frac{π}{2} + π n$ , for any integer $n$

Step 9.3

Consolidate $π n$ and $\frac{π}{2} + π n$ to $\frac{π n}{2}$ .

$x = \frac{π n}{2}, \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 9.4

Consolidate the answers.

$x = \frac{π n}{4}$ , for any integer $n$