Solve for ? 5sin(x)+1=3sin(x)

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Trigonometry

$5 \sin (x) + 1 = 3 \sin (x)$

Move all terms containing $\sin (x)$ to the left side of the equation.

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

$5 \sin (x) + 1 - 3 \sin (x) = 0$

Subtract $3 \sin (x)$ from $5 \sin (x)$ .

$2 \sin (x) + 1 = 0$

Subtract $1$ from both sides of the equation.

$2 \sin (x) = - 1$

Divide each term by $2$ and simplify.

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

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

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

Move the negative in front of the fraction.

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

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

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

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

$x = - \frac{π}{6}$

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.

$x = 2 π + \frac{π}{6} + π$

Simplify the expression to find the second solution.

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Simplify $2 π + \frac{π}{6} + π$ .

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To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = \frac{2 π}{1} \cdot \frac{6}{6} + \frac{π}{6} + π$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{2 π \cdot 6}{1 \cdot 6} + \frac{π}{6} + π$

Multiply $6$ by $1$ .

$x = \frac{2 π \cdot 6}{6} + \frac{π}{6} + π$

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 + π}{6} + π$

Simplify the numerator.

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Multiply $6$ by $2$ .

$x = \frac{12 π + π}{6} + π$

Add $12 π$ and $π$ .

$x = \frac{13 π}{6} + π$

To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = \frac{13 π}{6} + \frac{π}{1} \cdot \frac{6}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{13 π}{6} + \frac{π \cdot 6}{1 \cdot 6}$

Multiply $6$ by $1$ .

$x = \frac{13 π}{6} + \frac{π \cdot 6}{6}$

Combine the numerators over the common denominator.

$x = \frac{13 π + π \cdot 6}{6}$

Simplify the numerator.

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Move $6$ to the left of $π$ .

$x = \frac{13 π + 6 \cdot π}{6}$

Add $13 π$ and $6 π$ .

$x = \frac{19 π}{6}$

Subtract $2 π$ from $\frac{19 π}{6}$ .

$x = \frac{19 π}{6} - 2 π$

The resulting angle of $\frac{7 π}{6}$ is positive, less than $2 π$ , and coterminal with $\frac{19 π}{6}$ .

$x = \frac{7 π}{6}$

Find the period.

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

Solve the equation.

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

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Add $2 π$ to $- \frac{π}{6}$ to find the positive angle.

$- \frac{π}{6} + 2 π$

To write $\frac{2 π}{1}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{2 π}{1} \cdot \frac{6}{6} - \frac{π}{6}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{2 π \cdot 6}{1 \cdot 6} - \frac{π}{6}$

Multiply $6$ by $1$ .

$\frac{2 π \cdot 6}{6} - \frac{π}{6}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 6 - π}{6}$

Simplify the numerator.

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Multiply $6$ by $2$ .

$\frac{12 π - π}{6}$

Subtract $π$ from $12 π$ .

$\frac{11 π}{6}$

List the new angles.

$x = \frac{11 π}{6}$

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

$x = \frac{7 π}{6} + 2 π n, \frac{11 π}{6} + 2 π n$ , for any integer $n$

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