Trigonometry Examples

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

Step 1

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

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

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 $\sqrt{2 \sin (x) + 2}$ as ${(2 \sin (x) + 2)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(3 {(2 \sin (x) + 2)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $3 {(2 \sin (x) + 2)}^{\frac{1}{2}}$ .

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

Step 2.2.1.2

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

$9 {({(2 \sin (x) + 2)}^{\frac{1}{2}})}^{2} = {(- 1)}^{2}$

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

$9 {(2 \sin (x) + 2)}^{\frac{1}{2} \cdot 2} = {(- 1)}^{2}$

Step 2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$9 {(2 \sin (x) + 2)}^{\frac{1}{2} \cdot 2} = {(- 1)}^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$9 {(2 \sin (x) + 2)}^{1} = {(- 1)}^{2}$

Step 2.2.1.4

Simplify.

$9 (2 \sin (x) + 2) = {(- 1)}^{2}$

Step 2.2.1.5

Apply the distributive property.

$9 (2 \sin (x)) + 9 \cdot 2 = {(- 1)}^{2}$

Step 2.2.1.6

Multiply.

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Step 2.2.1.6.1

Multiply $2$ by $9$ .

$18 \sin (x) + 9 \cdot 2 = {(- 1)}^{2}$

Step 2.2.1.6.2

Multiply $9$ by $2$ .

$18 \sin (x) + 18 = {(- 1)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $- 1$ to the power of $2$ .

$18 \sin (x) + 18 = 1$

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 $18$ from both sides of the equation.

$18 \sin (x) = 1 - 18$

Step 3.1.2

Subtract $18$ from $1$ .

$18 \sin (x) = - 17$

Step 3.2

Divide each term in $18 \sin (x) = - 17$ by $18$ and simplify.

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Step 3.2.1

Divide each term in $18 \sin (x) = - 17$ by $18$ .

$\frac{18 \sin (x)}{18} = \frac{- 17}{18}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $18$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{18 \sin (x)}{18} = \frac{- 17}{18}$

Step 3.2.2.1.2

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

$\sin (x) = \frac{- 17}{18}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$\sin (x) = - \frac{17}{18}$

Step 3.3

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

$x = \arcsin (- \frac{17}{18})$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (- \frac{17}{18})$ .

$x = - 1.23590016$

Step 3.5

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.

$x = 2 (3.14159265) + 1.23590016 + 3.14159265$

Step 3.6

Simplify the expression to find the second solution.

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Step 3.6.1

Subtract $2 π$ from $2 (3.14159265) + 1.23590016 + 3.14159265$ .

$x = 2 (3.14159265) + 1.23590016 + 3.14159265 - 2 π$

Step 3.6.2

The resulting angle of $4.37749282$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 1.23590016 + 3.14159265$ .

$x = 4.37749282$

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

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

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Step 3.8.1

Add $2 π$ to $- 1.23590016$ to find the positive angle.

$- 1.23590016 + 2 π$

Step 3.8.2

Subtract $1.23590016$ from $2 π$ .

$5.04728513$

Step 3.8.3

List the new angles.

$x = 5.04728513$

Step 3.9

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

$x = 4.37749282 + 2 π n, 5.04728513 + 2 π n$ , for any integer $n$

Step 4

Exclude the solutions that do not make $3 \sqrt{2 \sin (x) + 2} = - 1$ true.

No solution