Trigonometry Examples

$3 \sin^{2} (x) - \sin (x) - 1 = 0$

Step 1

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

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

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

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

Step 4.1.2.2

Multiply $- 12$ by $- 1$ .

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

Step 4.1.3

Add $1$ and $12$ .

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

Step 4.2

Multiply $2$ by $3$ .

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

Step 5

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

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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 5.3

Change the $\pm$ to $+$ .

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

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

The final answer is the combination of both solutions.

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

Step 8

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

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

Step 9

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 10

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

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Step 10.1

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 10.2

Simplify the right side.

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Step 10.2.1

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

$x = 0.87507547$

Step 10.3

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.

$x = (3.14159265) - 0.87507547$

Step 10.4

Solve for $x$ .

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Step 10.4.1

Remove parentheses.

$x = 3.14159265 - 0.87507547$

Step 10.4.2

Remove parentheses.

$x = (3.14159265) - 0.87507547$

Step 10.4.3

Subtract $0.87507547$ from $3.14159265$ .

$x = 2.26651717$

Step 10.5

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

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Step 10.5.1

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

$\frac{2 π}{| b |}$

Step 10.5.2

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

$\frac{2 π}{| 1 |}$

Step 10.5.3

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

$\frac{2 π}{1}$

Step 10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.6

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

$x = 0.87507547 + 2 π n, 2.26651717 + 2 π n$ , for any integer $n$

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

$x = - 0.44921499$

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.

$x = (3.14159265) + 0.44921499$

Step 11.4

Solve for $x$ .

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Step 11.4.1

Remove parentheses.

$x = 3.14159265 + 0.44921499$

Step 11.4.2

Remove parentheses.

$x = (3.14159265) + 0.44921499$

Step 11.4.3

Add $3.14159265$ and $0.44921499$ .

$x = 3.59080764$

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

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

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Step 11.6.1

Add $2 π$ to $- 0.44921499$ to find the positive angle.

$- 0.44921499 + 2 π$

Step 11.6.2

Subtract $0.44921499$ from $2 π$ .

$5.83397031$

Step 11.6.3

List the new angles.

$x = 5.83397031$

Step 11.7

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

$x = 3.59080764 + 2 π n, 5.83397031 + 2 π n$ , for any integer $n$

Step 12

List all of the solutions.

$x = 0.87507547 + 2 π n, 2.26651717 + 2 π n, 3.59080764 + 2 π n, 5.83397031 + 2 π n$ , for any integer $n$