Trigonometry Examples

$12 \sin^{2} (x) - 6 \sin (x) = 4$

Step 1

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

$12 {(u)}^{2} - 6 u = 4$

Step 2

Subtract $4$ from both sides of the equation.

$12 u^{2} - 6 u - 4 = 0$

Step 3

Factor $2$ out of $12 u^{2} - 6 u - 4$ .

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Step 3.1

Factor $2$ out of $12 u^{2}$ .

$2 (6 u^{2}) - 6 u - 4 = 0$

Step 3.2

Factor $2$ out of $- 6 u$ .

$2 (6 u^{2}) + 2 (- 3 u) - 4 = 0$

Step 3.3

Factor $2$ out of $- 4$ .

$2 (6 u^{2}) + 2 (- 3 u) + 2 (- 2) = 0$

Step 3.4

Factor $2$ out of $2 (6 u^{2}) + 2 (- 3 u)$ .

$2 (6 u^{2} - 3 u) + 2 (- 2) = 0$

Step 3.5

Factor $2$ out of $2 (6 u^{2} - 3 u) + 2 (- 2)$ .

$2 (6 u^{2} - 3 u - 2) = 0$

Step 4

Divide each term in $2 (6 u^{2} - 3 u - 2) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (6 u^{2} - 3 u - 2) = 0$ by $2$ .

$\frac{2 (6 u^{2} - 3 u - 2)}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 (6 u^{2} - 3 u - 2)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $6 u^{2} - 3 u - 2$ by $1$ .

$6 u^{2} - 3 u - 2 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$6 u^{2} - 3 u - 2 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = 6$ , $b = - 3$ , and $c = - 2$ into the quadratic formula and solve for $u$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (6 \cdot - 2)}}{2 \cdot 6}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$u = \frac{3 \pm \sqrt{9 - 4 \cdot 6 \cdot - 2}}{2 \cdot 6}$

Step 7.1.2

Multiply $- 4 \cdot 6 \cdot - 2$ .

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Step 7.1.2.1

Multiply $- 4$ by $6$ .

$u = \frac{3 \pm \sqrt{9 - 24 \cdot - 2}}{2 \cdot 6}$

Step 7.1.2.2

Multiply $- 24$ by $- 2$ .

$u = \frac{3 \pm \sqrt{9 + 48}}{2 \cdot 6}$

Step 7.1.3

Add $9$ and $48$ .

$u = \frac{3 \pm \sqrt{57}}{2 \cdot 6}$

Step 7.2

Multiply $2$ by $6$ .

$u = \frac{3 \pm \sqrt{57}}{12}$

Step 8

The final answer is the combination of both solutions.

$u = \frac{3 + \sqrt{57}}{12}, \frac{3 - \sqrt{57}}{12}$

Step 9

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

$\sin (x) = \frac{3 + \sqrt{57}}{12}, \frac{3 - \sqrt{57}}{12}$

Step 10

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{3 + \sqrt{57}}{12}$

$\sin (x) = \frac{3 - \sqrt{57}}{12}$

Step 11

Solve for $x$ in $\sin (x) = \frac{3 + \sqrt{57}}{12}$ .

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Step 11.1

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

$x = \arcsin (\frac{3 + \sqrt{57}}{12})$

Step 11.2

Simplify the right side.

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Step 11.2.1

Evaluate $\arcsin (\frac{3 + \sqrt{57}}{12})$ .

$x = 1.0740816$

Step 11.3

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.

$x = (3.14159265) - 1.0740816$

Step 11.4

Solve for $x$ .

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Step 11.4.1

Remove parentheses.

$x = 3.14159265 - 1.0740816$

Step 11.4.2

Remove parentheses.

$x = (3.14159265) - 1.0740816$

Step 11.4.3

Subtract $1.0740816$ from $3.14159265$ .

$x = 2.06751104$

Step 11.5

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

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Step 11.5.1

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

$\frac{2 π}{| b |}$

Step 11.5.2

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

$\frac{2 π}{| 1 |}$

Step 11.5.3

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

$\frac{2 π}{1}$

Step 11.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 11.6

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

$x = 1.0740816 + 2 π n, 2.06751104 + 2 π n$ , for any integer $n$

Step 12

Solve for $x$ in $\sin (x) = \frac{3 - \sqrt{57}}{12}$ .

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Step 12.1

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

$x = \arcsin (\frac{3 - \sqrt{57}}{12})$

Step 12.2

Simplify the right side.

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Step 12.2.1

Evaluate $\arcsin (\frac{3 - \sqrt{57}}{12})$ .

$x = - 0.38888063$

Step 12.3

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.

$x = (3.14159265) + 0.38888063$

Step 12.4

Solve for $x$ .

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Step 12.4.1

Remove parentheses.

$x = 3.14159265 + 0.38888063$

Step 12.4.2

Remove parentheses.

$x = (3.14159265) + 0.38888063$

Step 12.4.3

Add $3.14159265$ and $0.38888063$ .

$x = 3.53047329$

Step 12.5

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

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Step 12.5.1

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

$\frac{2 π}{| b |}$

Step 12.5.2

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

$\frac{2 π}{| 1 |}$

Step 12.5.3

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

$\frac{2 π}{1}$

Step 12.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.6

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

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Step 12.6.1

Add $2 π$ to $- 0.38888063$ to find the positive angle.

$- 0.38888063 + 2 π$

Step 12.6.2

Subtract $0.38888063$ from $2 π$ .

$5.89430466$

Step 12.6.3

List the new angles.

$x = 5.89430466$

Step 12.7

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

$x = 3.53047329 + 2 π n, 5.89430466 + 2 π n$ , for any integer $n$

Step 13

List all of the solutions.

$x = 1.0740816 + 2 π n, 2.06751104 + 2 π n, 3.53047329 + 2 π n, 5.89430466 + 2 π n$ , for any integer $n$