Trigonometry Examples

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

Step 1

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

$3 {(u)}^{2} - (u) = 1$

Step 2

Subtract $1$ from both sides of the equation.

$3 u^{2} - u - 1 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 3$ , $b = - 1$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (3 \cdot - 1)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$u = \frac{1 \pm \sqrt{1 - 12 \cdot - 1}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 1$ .

$u = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot 3}$

Step 5.1.3

Add $1$ and $12$ .

$u = \frac{1 \pm \sqrt{13}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$u = \frac{1 \pm \sqrt{13}}{6}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Step 6.1.2

Multiply $- 4 \cdot 3 \cdot - 1$ .

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Step 6.1.2.1

Multiply $- 4$ by $3$ .

$u = \frac{1 \pm \sqrt{1 - 12 \cdot - 1}}{2 \cdot 3}$

Step 6.1.2.2

Multiply $- 12$ by $- 1$ .

$u = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot 3}$

Step 6.1.3

Add $1$ and $12$ .

$u = \frac{1 \pm \sqrt{13}}{2 \cdot 3}$

Step 6.2

Multiply $2$ by $3$ .

$u = \frac{1 \pm \sqrt{13}}{6}$

Step 6.3

Change the $\pm$ to $+$ .

$u = \frac{1 + \sqrt{13}}{6}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Step 7.1.2

Multiply $- 4 \cdot 3 \cdot - 1$ .

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Step 7.1.2.1

Multiply $- 4$ by $3$ .

$u = \frac{1 \pm \sqrt{1 - 12 \cdot - 1}}{2 \cdot 3}$

Step 7.1.2.2

Multiply $- 12$ by $- 1$ .

$u = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot 3}$

Step 7.1.3

Add $1$ and $12$ .

$u = \frac{1 \pm \sqrt{13}}{2 \cdot 3}$

Step 7.2

Multiply $2$ by $3$ .

$u = \frac{1 \pm \sqrt{13}}{6}$

Step 7.3

Change the $\pm$ to $-$ .

$u = \frac{1 - \sqrt{13}}{6}$

Step 8

The final answer is the combination of both solutions.

$u = \frac{1 + \sqrt{13}}{6}, \frac{1 - \sqrt{13}}{6}$

Step 9

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

$\sin (x) = \frac{1 + \sqrt{13}}{6}, \frac{1 - \sqrt{13}}{6}$

Step 10

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

$\sin (x) = \frac{1 + \sqrt{13}}{6}$

$\sin (x) = \frac{1 - \sqrt{13}}{6}$

Step 11

Solve for $x$ in $\sin (x) = \frac{1 + \sqrt{13}}{6}$ .

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Step 11.1

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

$x = \arcsin (\frac{1 + \sqrt{13}}{6})$

Step 11.2

Simplify the right side.

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Step 11.2.1

Evaluate $\arcsin (\frac{1 + \sqrt{13}}{6})$ .

$x = 50.13813146$

Step 11.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$x = 180 - 50.13813146$

Step 11.4

Subtract $50.13813146$ from $180$ .

$x = 129.86186853$

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 $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 11.5.2

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

$\frac{360}{| 1 |}$

Step 11.5.3

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

$\frac{360}{1}$

Step 11.5.4

Divide $360$ by $1$ .

$360$

Step 11.6

The period of the $\sin (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 50.13813146 + 360 n, 129.86186853 + 360 n$ , for any integer $n$

Step 12

Solve for $x$ in $\sin (x) = \frac{1 - \sqrt{13}}{6}$ .

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Step 12.1

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

$x = \arcsin (\frac{1 - \sqrt{13}}{6})$

Step 12.2

Simplify the right side.

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Step 12.2.1

Evaluate $\arcsin (\frac{1 - \sqrt{13}}{6})$ .

$x = - 25.73812338$

Step 12.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$x = 180 + 25.73812338$

Step 12.4

Add $180$ and $25.73812338$ .

$x = 205.73812338$

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 $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 12.5.2

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

$\frac{360}{| 1 |}$

Step 12.5.3

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

$\frac{360}{1}$

Step 12.5.4

Divide $360$ by $1$ .

$360$

Step 12.6

Add $360$ to every negative angle to get positive angles.

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Step 12.6.1

Add $360$ to $- 25.73812338$ to find the positive angle.

$- 25.73812338 + 360$

Step 12.6.2

Subtract $25.73812338$ from $360$ .

$334.26187661$

Step 12.6.3

List the new angles.

$x = 334.26187661$

Step 12.7

The period of the $\sin (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 205.73812338 + 360 n, 334.26187661 + 360 n$ , for any integer $n$

Step 13

List all of the solutions.

$x = 50.13813146 + 360 n, 129.86186853 + 360 n, 205.73812338 + 360 n, 334.26187661 + 360 n$ , for any integer $n$