Trigonometry Examples

$f (x) = 2 \sin (x) + \cos (2 x)$ , $f (x) = \frac{π}{6}$

Step 1

Substitute $\frac{π}{6}$ for $f (x)$ .

$\frac{π}{6} = 2 \sin (x) + \cos (2 x)$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $2 \sin (x) + \cos (2 x) = \frac{π}{6}$ .

$2 \sin (x) + \cos (2 x) = \frac{π}{6}$

Step 2.2

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$2 \sin (x) + 1 - 2 \sin^{2} (x) = \frac{π}{6}$

Step 2.3

Subtract $1$ from both sides of the equation.

$2 \sin (x) - 2 \sin^{2} (x) = \frac{π}{6} - 1$

Step 2.4

Solve the equation for $x$ .

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Step 2.4.1

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

$2 (u) - 2 {(u)}^{2} = \frac{π}{6} - 1$

Step 2.4.2

Move all the expressions to the left side of the equation.

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Step 2.4.2.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$2 u - 2 u^{2} - \frac{π}{6} = - 1$

Step 2.4.2.2

Add $1$ to both sides of the equation.

$2 u - 2 u^{2} - \frac{π}{6} + 1 = 0$

Step 2.4.3

Multiply through by the least common denominator $6$ , then simplify.

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Step 2.4.3.1

Apply the distributive property.

$6 (2 u) + 6 (- 2 u^{2}) + 6 (- \frac{π}{6}) + 6 \cdot 1 = 0$

Step 2.4.3.2

Simplify.

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Step 2.4.3.2.1

Multiply $2$ by $6$ .

$12 u + 6 (- 2 u^{2}) + 6 (- \frac{π}{6}) + 6 \cdot 1 = 0$

Step 2.4.3.2.2

Multiply $- 2$ by $6$ .

$12 u - 12 u^{2} + 6 (- \frac{π}{6}) + 6 \cdot 1 = 0$

Step 2.4.3.2.3

Cancel the common factor of $6$ .

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Step 2.4.3.2.3.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$12 u - 12 u^{2} + 6 (\frac{- π}{6}) + 6 \cdot 1 = 0$

Step 2.4.3.2.3.2

Cancel the common factor.

$12 u - 12 u^{2} + 6 (\frac{- π}{6}) + 6 \cdot 1 = 0$

Step 2.4.3.2.3.3

Rewrite the expression.

$12 u - 12 u^{2} - π + 6 \cdot 1 = 0$

Step 2.4.3.2.4

Multiply $6$ by $1$ .

$12 u - 12 u^{2} - π + 6 = 0$

Step 2.4.3.3

Reorder $12 u$ and $- 12 u^{2}$ .

$- 12 u^{2} + 12 u - π + 6 = 0$

Step 2.4.4

Use the quadratic formula to find the solutions.

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

Step 2.4.5

Substitute the values $a = - 12$ , $b = 12$ , and $c = - π + 6$ into the quadratic formula and solve for $u$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (- 12 \cdot (- π + 6))}}{2 \cdot - 12}$

Step 2.4.6

Simplify.

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Step 2.4.6.1

Simplify the numerator.

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Step 2.4.6.1.1

Raise $12$ to the power of $2$ .

$u = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 12 \cdot (- π + 6)}}{2 \cdot - 12}$

Step 2.4.6.1.2

Multiply $- 4$ by $- 12$ .

$u = \frac{- 12 \pm \sqrt{144 + 48 \cdot (- π + 6)}}{2 \cdot - 12}$

Step 2.4.6.1.3

Apply the distributive property.

$u = \frac{- 12 \pm \sqrt{144 + 48 (- π) + 48 \cdot 6}}{2 \cdot - 12}$

Step 2.4.6.1.4

Multiply $- 1$ by $48$ .

$u = \frac{- 12 \pm \sqrt{144 - 48 π + 48 \cdot 6}}{2 \cdot - 12}$

Step 2.4.6.1.5

Multiply $48$ by $6$ .

$u = \frac{- 12 \pm \sqrt{144 - 48 π + 288}}{2 \cdot - 12}$

Step 2.4.6.1.6

Add $144$ and $288$ .

$u = \frac{- 12 \pm \sqrt{432 - 48 π}}{2 \cdot - 12}$

Step 2.4.6.2

Multiply $2$ by $- 12$ .

$u = \frac{- 12 \pm \sqrt{432 - 48 π}}{- 24}$

Step 2.4.6.3

Simplify $\frac{- 12 \pm \sqrt{432 - 48 π}}{- 24}$ .

$u = \frac{12 \pm \sqrt{432 - 48 π}}{24}$

Step 2.4.7

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

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Step 2.4.7.1

Simplify the numerator.

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Step 2.4.7.1.1

Raise $12$ to the power of $2$ .

$u = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 12 \cdot (- π + 6)}}{2 \cdot - 12}$

Step 2.4.7.1.2

Multiply $- 4$ by $- 12$ .

$u = \frac{- 12 \pm \sqrt{144 + 48 \cdot (- π + 6)}}{2 \cdot - 12}$

Step 2.4.7.1.3

Apply the distributive property.

$u = \frac{- 12 \pm \sqrt{144 + 48 (- π) + 48 \cdot 6}}{2 \cdot - 12}$

Step 2.4.7.1.4

Multiply $- 1$ by $48$ .

$u = \frac{- 12 \pm \sqrt{144 - 48 π + 48 \cdot 6}}{2 \cdot - 12}$

Step 2.4.7.1.5

Multiply $48$ by $6$ .

$u = \frac{- 12 \pm \sqrt{144 - 48 π + 288}}{2 \cdot - 12}$

Step 2.4.7.1.6

Add $144$ and $288$ .

$u = \frac{- 12 \pm \sqrt{432 - 48 π}}{2 \cdot - 12}$

Step 2.4.7.2

Multiply $2$ by $- 12$ .

$u = \frac{- 12 \pm \sqrt{432 - 48 π}}{- 24}$

Step 2.4.7.3

Simplify $\frac{- 12 \pm \sqrt{432 - 48 π}}{- 24}$ .

$u = \frac{12 \pm \sqrt{432 - 48 π}}{24}$

Step 2.4.7.4

Change the $\pm$ to $+$ .

$u = \frac{12 + \sqrt{432 - 48 π}}{24}$

Step 2.4.8

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

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Step 2.4.8.1

Simplify the numerator.

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Step 2.4.8.1.1

Raise $12$ to the power of $2$ .

$u = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 12 \cdot (- π + 6)}}{2 \cdot - 12}$

Step 2.4.8.1.2

Multiply $- 4$ by $- 12$ .

$u = \frac{- 12 \pm \sqrt{144 + 48 \cdot (- π + 6)}}{2 \cdot - 12}$

Step 2.4.8.1.3

Apply the distributive property.

$u = \frac{- 12 \pm \sqrt{144 + 48 (- π) + 48 \cdot 6}}{2 \cdot - 12}$

Step 2.4.8.1.4

Multiply $- 1$ by $48$ .

$u = \frac{- 12 \pm \sqrt{144 - 48 π + 48 \cdot 6}}{2 \cdot - 12}$

Step 2.4.8.1.5

Multiply $48$ by $6$ .

$u = \frac{- 12 \pm \sqrt{144 - 48 π + 288}}{2 \cdot - 12}$

Step 2.4.8.1.6

Add $144$ and $288$ .

$u = \frac{- 12 \pm \sqrt{432 - 48 π}}{2 \cdot - 12}$

Step 2.4.8.2

Multiply $2$ by $- 12$ .

$u = \frac{- 12 \pm \sqrt{432 - 48 π}}{- 24}$

Step 2.4.8.3

Simplify $\frac{- 12 \pm \sqrt{432 - 48 π}}{- 24}$ .

$u = \frac{12 \pm \sqrt{432 - 48 π}}{24}$

Step 2.4.8.4

Change the $\pm$ to $-$ .

$u = \frac{12 - \sqrt{432 - 48 π}}{24}$

Step 2.4.9

The final answer is the combination of both solutions.

$u = \frac{12 + \sqrt{432 - 48 π}}{24}, \frac{12 - \sqrt{432 - 48 π}}{24}$

Step 2.4.10

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

$\sin (x) = \frac{12 + \sqrt{432 - 48 π}}{24}, \frac{12 - \sqrt{432 - 48 π}}{24}$

Step 2.4.11

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

$\sin (x) = \frac{12 + \sqrt{432 - 48 π}}{24}$

$\sin (x) = \frac{12 - \sqrt{432 - 48 π}}{24}$

Step 2.4.12

Solve for $x$ in $\sin (x) = \frac{12 + \sqrt{432 - 48 π}}{24}$ .

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Step 2.4.12.1

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

$x = \arcsin (\frac{12 + \sqrt{432 - 48 π}}{24})$

Step 2.4.12.2

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.

$x = π - \arcsin (\frac{12 + \sqrt{432 - 48 π}}{24})$

Step 2.4.12.3

Remove parentheses.

$x = π - \arcsin (\frac{12 + \sqrt{432 - 48 π}}{24})$

Step 2.4.12.4

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

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Step 2.4.12.4.1

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

$\frac{2 π}{| b |}$

Step 2.4.12.4.2

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

$\frac{2 π}{| 1 |}$

Step 2.4.12.4.3

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

$\frac{2 π}{1}$

Step 2.4.12.4.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.4.12.5

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

$x = \arcsin (\frac{12 + \sqrt{432 - 48 π}}{24}) + 2 π n, π - \arcsin (\frac{12 + \sqrt{432 - 48 π}}{24}) + 2 π n$ , for any integer $n$

Step 2.4.13

Solve for $x$ in $\sin (x) = \frac{12 - \sqrt{432 - 48 π}}{24}$ .

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Step 2.4.13.1

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

$x = \arcsin (\frac{12 - \sqrt{432 - 48 π}}{24})$

Step 2.4.13.2

Simplify the right side.

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Step 2.4.13.2.1

Evaluate $\arcsin (\frac{12 - \sqrt{432 - 48 π}}{24})$ .

$x = - 0.2000451$

Step 2.4.13.3

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.

$x = (3.14159265) + 0.2000451$

Step 2.4.13.4

Solve for $x$ .

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Step 2.4.13.4.1

Remove parentheses.

$x = 3.14159265 + 0.2000451$

Step 2.4.13.4.2

Remove parentheses.

$x = (3.14159265) + 0.2000451$

Step 2.4.13.4.3

Add $3.14159265$ and $0.2000451$ .

$x = 3.34163776$

Step 2.4.13.5

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

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Step 2.4.13.5.1

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

$\frac{2 π}{| b |}$

Step 2.4.13.5.2

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

$\frac{2 π}{| 1 |}$

Step 2.4.13.5.3

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

$\frac{2 π}{1}$

Step 2.4.13.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.4.13.6

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

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Step 2.4.13.6.1

Add $2 π$ to $- 0.2000451$ to find the positive angle.

$- 0.2000451 + 2 π$

Step 2.4.13.6.2

Subtract $0.2000451$ from $2 π$ .

$6.08314019$

Step 2.4.13.6.3

List the new angles.

$x = 6.08314019$

Step 2.4.13.7

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

$x = 3.34163776 + 2 π n, 6.08314019 + 2 π n$ , for any integer $n$

Step 2.4.14

List all of the solutions.

$x = \arcsin (\frac{12 + \sqrt{432 - 48 π}}{24}) + 2 π n, π - \arcsin (\frac{12 + \sqrt{432 - 48 π}}{24}) + 2 π n, 3.34163776 + 2 π n, 6.08314019 + 2 π n$ , for any integer $n$