Precalculus Examples

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

Step 1

Subtract $3 \sin (x)$ from both sides of the equation.

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

Step 2

Simplify $3 \sin (x) \cos^{2} (x) - 3 \sin (x)$ .

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Step 2.1

Reorder $3 \sin (x) \cos^{2} (x)$ and $- 3 \sin (x)$ .

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

Step 2.2

Factor $- 3 \sin (x)$ out of $- 3 \sin (x)$ .

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

Step 2.3

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

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

Step 2.4

Factor $- 3 \sin (x)$ out of $- 3 \sin (x) (1) - 3 \sin (x) (- 1 \cos^{2} (x))$ .

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

Step 2.5

Rewrite $- 1 \cos^{2} (x)$ as $- \cos^{2} (x)$ .

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

Step 2.6

Apply pythagorean identity.

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

Step 2.7

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 2.7.1

Move $\sin^{2} (x)$ .

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

Step 2.7.2

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

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Step 2.7.2.1

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

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

Step 2.7.2.2

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

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

Step 2.7.3

Add $2$ and $1$ .

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

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $- 3 \sin^{3} (x) = 0$ by $- 3$ and simplify.

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Step 3.1.1

Divide each term in $- 3 \sin^{3} (x) = 0$ by $- 3$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 3$ .

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Step 3.1.2.1.1

Cancel the common factor.

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

Step 3.1.2.1.2

Divide $\sin^{3} (x)$ by $1$ .

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $- 3$ .

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

Step 3.2

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

$\sin (x) = \sqrt[3]{0}$

Step 3.3

Simplify $\sqrt[3]{0}$ .

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Step 3.3.1

Rewrite $0$ as $0^{3}$ .

$\sin (x) = \sqrt[3]{0^{3}}$

Step 3.3.2

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

$\sin (x) = 0$

Step 3.4

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

$x = \arcsin (0)$

Step 3.5

Simplify the right side.

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Step 3.5.1

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

$x = 0$

Step 3.6

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 3.7

Subtract $0$ from $π$ .

$x = π$

Step 3.8

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

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Step 3.8.1

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

$\frac{2 π}{| b |}$

Step 3.8.2

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

$\frac{2 π}{| 1 |}$

Step 3.8.3

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

$\frac{2 π}{1}$

Step 3.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.9

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 4

Consolidate the answers.

$x = π n$ , for any integer $n$