Trigonometry Examples

$9 \sin^{2} (x) + 18 \sin (x) + 9 = 0$

Step 1

Substitute $u$ for $\sin (x)$ .

$9 {(u)}^{2} + 18 (u) + 9 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $9$ out of $9 u^{2} + 18 u + 9$ .

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Step 2.1.1

Factor $9$ out of $9 u^{2}$ .

$9 u^{2} + 18 u + 9 = 0$

Step 2.1.2

Factor $9$ out of $18 u$ .

$9 u^{2} + 9 (2 u) + 9 = 0$

Step 2.1.3

Factor $9$ out of $9$ .

$9 u^{2} + 9 (2 u) + 9 (1) = 0$

Step 2.1.4

Factor $9$ out of $9 u^{2} + 9 (2 u)$ .

$9 (u^{2} + 2 u) + 9 (1) = 0$

Step 2.1.5

Factor $9$ out of $9 (u^{2} + 2 u) + 9 (1)$ .

$9 (u^{2} + 2 u + 1) = 0$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

Rewrite $1$ as $1^{2}$ .

$9 (u^{2} + 2 u + 1^{2}) = 0$

Step 2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 2.2.3

Rewrite the polynomial.

$9 (u^{2} + 2 \cdot u \cdot 1 + 1^{2}) = 0$

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 1$ .

$9 {(u + 1)}^{2} = 0$

Step 3

Divide each term in $9 {(u + 1)}^{2} = 0$ by $9$ and simplify.

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Step 3.1

Divide each term in $9 {(u + 1)}^{2} = 0$ by $9$ .

$\frac{9 {(u + 1)}^{2}}{9} = \frac{0}{9}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $9$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{9 {(u + 1)}^{2}}{9} = \frac{0}{9}$

Step 3.2.1.2

Divide ${(u + 1)}^{2}$ by $1$ .

${(u + 1)}^{2} = \frac{0}{9}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $9$ .

${(u + 1)}^{2} = 0$

Step 4

Set the $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 5

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 6

Substitute $\sin (x)$ for $u$ .

$\sin (x) = - 1$

Step 7

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

$x = \arcsin (- 1)$

Step 8

Simplify the right side.

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Step 8.1

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

$x = - \frac{π}{2}$

Step 9

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.

$x = 2 π + \frac{π}{2} + π$

Step 10

Simplify the expression to find the second solution.

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Step 10.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 10.2

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

$x = \frac{3 π}{2}$

Step 11

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

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Step 11.1

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

$\frac{2 π}{| b |}$

Step 11.2

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

$\frac{2 π}{| 1 |}$

Step 11.3

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

$\frac{2 π}{1}$

Step 11.4

Divide $2 π$ by $1$ .

$2 π$

Step 12

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

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Step 12.1

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

$- \frac{π}{2} + 2 π$

Step 12.2

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

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 12.3

Combine fractions.

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Step 12.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 12.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 12.4

Simplify the numerator.

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Step 12.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 12.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 12.5

List the new angles.

$x = \frac{3 π}{2}$

Step 13

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 $n$

Step 14

Consolidate the answers.

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