Solve for x 3sin(x)^2+sin(x)-2=0

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Trigonometry

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

Factor the left side of the equation.

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Let $u = \sin (x)$ . Substitute $u$ for all occurrences of $\sin (x)$ .

$3 u^{2} + u - 2$

Factor by grouping.

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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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = 1$ .

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Multiply by $1$ .

$3 u^{2} + 1 u - 2$

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

$3 u^{2} + (- 2 + 3) u - 2$

Apply the distributive property.

$3 u^{2} - 2 u + 3 u - 2$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$(3 u^{2} - 2 u) + 3 u - 2$

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

$u (3 u - 2) + 1 (3 u - 2)$

Factor the polynomial by factoring out the greatest common factor, $3 u - 2$ .

$(3 u - 2) (u + 1)$

Replace all occurrences of $u$ with $\sin (x)$ .

$(3 \sin (x) - 2) (\sin (x) + 1)$

Replace the left side with the factored expression.

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

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

$3 \sin (x) - 2 = 0$

$\sin (x) + 1 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$3 \sin (x) - 2 = 0$

Add $2$ to both sides of the equation.

$3 \sin (x) = 2$

Divide each term by $3$ and simplify.

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Divide each term in $3 \sin (x) = 2$ by $3$ .

$\frac{3 \sin (x)}{3} = \frac{2}{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 \sin (x)}{3} = \frac{2}{3}$

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

$\sin (x) = \frac{2}{3}$

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

$x = \arcsin (\frac{2}{3})$

Evaluate $\arcsin (\frac{2}{3})$ .

$x = 0.72972765$

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.

$x = (3.14159265) - 0.72972765$

Simplify the expression to find the second solution.

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Remove the parentheses around the expression $3.14159265$ .

$x = 3.14159265 - 0.72972765$

Subtract $0.72972765$ from $3.14159265$ .

$x = 2.41186499$

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 $1$ .

$2 π$

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

$x = 0.72972765 + 2 π n, 2.41186499 + 2 π n$ , for any integer $n$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$\sin (x) + 1 = 0$

Subtract $1$ from both sides of the equation.

$\sin (x) = - 1$

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

$x = \arcsin (- 1)$

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

$x = - \frac{π}{2}$

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.

$x = 2 π + \frac{π}{2} + π$

Simplify the expression to find the second solution.

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Simplify $2 π + \frac{π}{2} + π$ .

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

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

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Multiply $2$ by $1$ .

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

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 + π}{2} + π$

Simplify the numerator.

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Multiply $2$ by $2$ .

$x = \frac{4 π + π}{2} + π$

Add $4 π$ and $π$ .

$x = \frac{5 π}{2} + π$

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

$x = \frac{5 π}{2} + \frac{π}{1} \cdot \frac{2}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$x = \frac{5 π}{2} + \frac{π \cdot 2}{1 \cdot 2}$

Multiply $2$ by $1$ .

$x = \frac{5 π}{2} + \frac{π \cdot 2}{2}$

Combine the numerators over the common denominator.

$x = \frac{5 π + π \cdot 2}{2}$

Simplify the numerator.

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Move $2$ to the left of $π$ .

$x = \frac{5 π + 2 \cdot π}{2}$

Add $5 π$ and $2 π$ .

$x = \frac{7 π}{2}$

Subtract $2 π$ from $\frac{7 π}{2}$ .

$x = \frac{7 π}{2} - 2 π$

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $\frac{7 π}{2}$ .

$x = \frac{3 π}{2}$

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

$- \frac{π}{2} + 2 π$

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

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

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Multiply $2$ by $1$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

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Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

List the new angles.

$x = \frac{3 π}{2}$

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 $n$

The final solution is all the values that make $(3 \sin (x) - 2) (\sin (x) + 1) = 0$ true.

$x = 0.72972765 + 2 π n, 2.41186499 + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

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