Trigonometry Examples

$8 \sin^{2} (x) - 4 \sin (x) = 2$

Step 1

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

$8 {(u)}^{2} - 4 u = 2$

Step 2

Subtract $2$ from both sides of the equation.

$8 u^{2} - 4 u - 2 = 0$

Step 3

Factor $2$ out of $8 u^{2} - 4 u - 2$ .

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Step 3.1

Factor $2$ out of $8 u^{2}$ .

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

Step 3.2

Factor $2$ out of $- 4 u$ .

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

Step 3.3

Factor $2$ out of $- 2$ .

$2 (4 u^{2}) + 2 (- 2 u) + 2 (- 1) = 0$

Step 3.4

Factor $2$ out of $2 (4 u^{2}) + 2 (- 2 u)$ .

$2 (4 u^{2} - 2 u) + 2 (- 1) = 0$

Step 3.5

Factor $2$ out of $2 (4 u^{2} - 2 u) + 2 (- 1)$ .

$2 (4 u^{2} - 2 u - 1) = 0$

Step 4

Divide each term in $2 (4 u^{2} - 2 u - 1) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (4 u^{2} - 2 u - 1) = 0$ by $2$ .

$\frac{2 (4 u^{2} - 2 u - 1)}{2} = \frac{0}{2}$

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.

$\frac{2 (4 u^{2} - 2 u - 1)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $4 u^{2} - 2 u - 1$ by $1$ .

$4 u^{2} - 2 u - 1 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$4 u^{2} - 2 u - 1 = 0$

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 $a = 4$ , $b = - 2$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

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

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$u = \frac{2 \pm \sqrt{4 - 16 \cdot - 1}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 1$ .

$u = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 4}$

Step 7.1.3

Add $4$ and $16$ .

$u = \frac{2 \pm \sqrt{20}}{2 \cdot 4}$

Step 7.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 7.1.4.1

Factor $4$ out of $20$ .

$u = \frac{2 \pm \sqrt{4 (5)}}{2 \cdot 4}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$u = \frac{2 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 4}$

Step 7.1.5

Pull terms out from under the radical.

$u = \frac{2 \pm 2 \sqrt{5}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$u = \frac{2 \pm 2 \sqrt{5}}{8}$

Step 7.3

Simplify $\frac{2 \pm 2 \sqrt{5}}{8}$ .

$u = \frac{1 \pm \sqrt{5}}{4}$

Step 8

The final answer is the combination of both solutions.

$u = \frac{1 + \sqrt{5}}{4}, \frac{1 - \sqrt{5}}{4}$

Step 9

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

$\sin (x) = \frac{1 + \sqrt{5}}{4}, \frac{1 - \sqrt{5}}{4}$

Step 10

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

$\sin (x) = \frac{1 + \sqrt{5}}{4}$

$\sin (x) = \frac{1 - \sqrt{5}}{4}$

Step 11

Solve for $x$ in $\sin (x) = \frac{1 + \sqrt{5}}{4}$ .

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Step 11.1

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

$x = \arcsin (\frac{1 + \sqrt{5}}{4})$

Step 11.2

Simplify the right side.

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Step 11.2.1

Evaluate $\arcsin (\frac{1 + \sqrt{5}}{4})$ .

$x = \frac{3 π}{10}$

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.

$x = π - \frac{3 π}{10}$

Step 11.4

Simplify $π - \frac{3 π}{10}$ .

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Step 11.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$x = π \cdot \frac{10}{10} - \frac{3 π}{10}$

Step 11.4.2

Combine fractions.

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Step 11.4.2.1

Combine $π$ and $\frac{10}{10}$ .

$x = \frac{π \cdot 10}{10} - \frac{3 π}{10}$

Step 11.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 10 - 3 π}{10}$

Step 11.4.3

Simplify the numerator.

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Step 11.4.3.1

Move $10$ to the left of $π$ .

$x = \frac{10 \cdot π - 3 π}{10}$

Step 11.4.3.2

Subtract $3 π$ from $10 π$ .

$x = \frac{7 π}{10}$

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.

$x = \frac{3 π}{10} + 2 π n, \frac{7 π}{10} + 2 π n$ , for any integer $n$

Step 12

Solve for $x$ in $\sin (x) = \frac{1 - \sqrt{5}}{4}$ .

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Step 12.1

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

$x = \arcsin (\frac{1 - \sqrt{5}}{4})$

Step 12.2

Simplify the right side.

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Step 12.2.1

Evaluate $\arcsin (\frac{1 - \sqrt{5}}{4})$ .

$x = - \frac{π}{10}$

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.

$x = π + \frac{π}{10}$

Step 12.4

Simplify $π + \frac{π}{10}$ .

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Step 12.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$x = π \cdot \frac{10}{10} + \frac{π}{10}$

Step 12.4.2

Combine fractions.

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Step 12.4.2.1

Combine $π$ and $\frac{10}{10}$ .

$x = \frac{π \cdot 10}{10} + \frac{π}{10}$

Step 12.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 10 + π}{10}$

Step 12.4.3

Simplify the numerator.

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Step 12.4.3.1

Move $10$ to the left of $π$ .

$x = \frac{10 \cdot π + π}{10}$

Step 12.4.3.2

Add $10 π$ and $π$ .

$x = \frac{11 π}{10}$

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 $- \frac{π}{10}$ to find the positive angle.

$- \frac{π}{10} + 2 π$

Step 12.6.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$2 π \cdot \frac{10}{10} - \frac{π}{10}$

Step 12.6.3

Combine fractions.

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Step 12.6.3.1

Combine $2 π$ and $\frac{10}{10}$ .

$\frac{2 π \cdot 10}{10} - \frac{π}{10}$

Step 12.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 10 - π}{10}$

Step 12.6.4

Simplify the numerator.

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Step 12.6.4.1

Multiply $10$ by $2$ .

$\frac{20 π - π}{10}$

Step 12.6.4.2

Subtract $π$ from $20 π$ .

$\frac{19 π}{10}$

Step 12.6.5

List the new angles.

$x = \frac{19 π}{10}$

Step 12.7

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

$x = \frac{11 π}{10} + 2 π n, \frac{19 π}{10} + 2 π n$ , for any integer $n$

Step 13

List all of the solutions.

$x = \frac{3 π}{10} + 2 π n, \frac{7 π}{10} + 2 π n, \frac{11 π}{10} + 2 π n, \frac{19 π}{10} + 2 π n$ , for any integer $n$