Trigonometry Examples

$\sqrt{4 \sin (x) + 7} = 3$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 \sin (x) + 7}}^{2} = 3^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 \sin (x) + 7}$ as ${(4 \sin (x) + 7)}^{\frac{1}{2}}$ .

${({(4 \sin (x) + 7)}^{\frac{1}{2}})}^{2} = 3^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(4 \sin (x) + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(4 \sin (x) + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(4 \sin (x) + 7)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(4 \sin (x) + 7)}^{\frac{1}{2} \cdot 2} = 3^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(4 \sin (x) + 7)}^{1} = 3^{2}$

Step 2.2.1.2

Simplify.

$4 \sin (x) + 7 = 3^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $3$ to the power of $2$ .

$4 \sin (x) + 7 = 9$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms not containing $\sin (x)$ to the right side of the equation.

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Step 3.1.1

Subtract $7$ from both sides of the equation.

$4 \sin (x) = 9 - 7$

Step 3.1.2

Subtract $7$ from $9$ .

$4 \sin (x) = 2$

Step 3.2

Divide each term in $4 \sin (x) = 2$ by $4$ and simplify.

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Step 3.2.1

Divide each term in $4 \sin (x) = 2$ by $4$ .

$\frac{4 \sin (x)}{4} = \frac{2}{4}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{4 \sin (x)}{4} = \frac{2}{4}$

Step 3.2.2.1.2

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

$\sin (x) = \frac{2}{4}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $2$ and $4$ .

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Step 3.2.3.1.1

Factor $2$ out of $2$ .

$\sin (x) = \frac{2 (1)}{4}$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $2$ out of $4$ .

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

Step 3.2.3.1.2.2

Cancel the common factor.

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

Step 3.2.3.1.2.3

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Simplify the right side.

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Step 3.4.1

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

$x = \frac{π}{6}$

Step 3.5

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.

$x = π - \frac{π}{6}$

Step 3.6

Simplify $π - \frac{π}{6}$ .

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Step 3.6.1

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

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

Step 3.6.2

Combine fractions.

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Step 3.6.2.1

Combine $π$ and $\frac{6}{6}$ .

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

Step 3.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 3.6.3

Simplify the numerator.

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Step 3.6.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 3.6.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 3.7

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

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Step 3.7.1

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

$\frac{2 π}{| b |}$

Step 3.7.2

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

$\frac{2 π}{| 1 |}$

Step 3.7.3

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

$\frac{2 π}{1}$

Step 3.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.8

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

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