Trigonometry Examples

$f (x) = 2 \sin (x) + \cos (2 x)$ , $f (x) = 2$

Step 1

Substitute $2$ for $f (x)$ .

$2 = 2 \sin (x) + \cos (2 x)$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $2 \sin (x) + \cos (2 x) = 2$ .

$2 \sin (x) + \cos (2 x) = 2$

Step 2.2

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

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

Step 2.3

Subtract $1$ from both sides of the equation.

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

Step 2.4

Simplify the right side.

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Step 2.4.1

Subtract $1$ from $2$ .

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

Step 2.5

Solve the equation for $x$ .

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Step 2.5.1

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

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

Step 2.5.2

Subtract $1$ from both sides of the equation.

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

Step 2.5.3

Use the quadratic formula to find the solutions.

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

Step 2.5.4

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

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

Step 2.5.5

Simplify.

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Step 2.5.5.1

Simplify the numerator.

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Step 2.5.5.1.1

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

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

Step 2.5.5.1.2

Multiply $- 4 \cdot - 2 \cdot - 1$ .

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Step 2.5.5.1.2.1

Multiply $- 4$ by $- 2$ .

$u = \frac{- 2 \pm \sqrt{4 + 8 \cdot - 1}}{2 \cdot - 2}$

Step 2.5.5.1.2.2

Multiply $8$ by $- 1$ .

$u = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot - 2}$

Step 2.5.5.1.3

Subtract $8$ from $4$ .

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

Step 2.5.5.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.5.5.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.5.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 2.5.5.1.7

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

$u = \frac{- 2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot - 2}$

Step 2.5.5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$u = \frac{- 2 \pm i \cdot 2}{2 \cdot - 2}$

Step 2.5.5.1.9

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i}{2 \cdot - 2}$

Step 2.5.5.2

Multiply $2$ by $- 2$ .

$u = \frac{- 2 \pm 2 i}{- 4}$

Step 2.5.5.3

Simplify $\frac{- 2 \pm 2 i}{- 4}$ .

$u = \frac{1 \pm i}{2}$

Step 2.5.6

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

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Step 2.5.6.1

Simplify the numerator.

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Step 2.5.6.1.1

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

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

Step 2.5.6.1.2

Multiply $- 4 \cdot - 2 \cdot - 1$ .

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Step 2.5.6.1.2.1

Multiply $- 4$ by $- 2$ .

$u = \frac{- 2 \pm \sqrt{4 + 8 \cdot - 1}}{2 \cdot - 2}$

Step 2.5.6.1.2.2

Multiply $8$ by $- 1$ .

$u = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot - 2}$

Step 2.5.6.1.3

Subtract $8$ from $4$ .

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

Step 2.5.6.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.5.6.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.5.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 2.5.6.1.7

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

$u = \frac{- 2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot - 2}$

Step 2.5.6.1.8

Pull terms out from under the radical, assuming positive real numbers.

$u = \frac{- 2 \pm i \cdot 2}{2 \cdot - 2}$

Step 2.5.6.1.9

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i}{2 \cdot - 2}$

Step 2.5.6.2

Multiply $2$ by $- 2$ .

$u = \frac{- 2 \pm 2 i}{- 4}$

Step 2.5.6.3

Simplify $\frac{- 2 \pm 2 i}{- 4}$ .

$u = \frac{1 \pm i}{2}$

Step 2.5.6.4

Change the $\pm$ to $+$ .

$u = \frac{1 + i}{2}$

Step 2.5.6.5

Split the fraction $\frac{1 + i}{2}$ into two fractions.

$u = \frac{1}{2} + \frac{i}{2}$

Step 2.5.7

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

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Step 2.5.7.1

Simplify the numerator.

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Step 2.5.7.1.1

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

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

Step 2.5.7.1.2

Multiply $- 4 \cdot - 2 \cdot - 1$ .

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Step 2.5.7.1.2.1

Multiply $- 4$ by $- 2$ .

$u = \frac{- 2 \pm \sqrt{4 + 8 \cdot - 1}}{2 \cdot - 2}$

Step 2.5.7.1.2.2

Multiply $8$ by $- 1$ .

$u = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot - 2}$

Step 2.5.7.1.3

Subtract $8$ from $4$ .

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

Step 2.5.7.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.5.7.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.5.7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 2.5.7.1.7

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

$u = \frac{- 2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot - 2}$

Step 2.5.7.1.8

Pull terms out from under the radical, assuming positive real numbers.

$u = \frac{- 2 \pm i \cdot 2}{2 \cdot - 2}$

Step 2.5.7.1.9

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i}{2 \cdot - 2}$

Step 2.5.7.2

Multiply $2$ by $- 2$ .

$u = \frac{- 2 \pm 2 i}{- 4}$

Step 2.5.7.3

Simplify $\frac{- 2 \pm 2 i}{- 4}$ .

$u = \frac{1 \pm i}{2}$

Step 2.5.7.4

Change the $\pm$ to $-$ .

$u = \frac{1 - i}{2}$

Step 2.5.7.5

Split the fraction $\frac{1 - i}{2}$ into two fractions.

$u = \frac{1}{2} + \frac{- i}{2}$

Step 2.5.7.6

Move the negative in front of the fraction.

$u = \frac{1}{2} - \frac{i}{2}$

Step 2.5.8

The final answer is the combination of both solutions.

$u = \frac{1}{2} + \frac{i}{2}, \frac{1}{2} - \frac{i}{2}$

Step 2.5.9

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

$\sin (x) = \frac{1}{2} + \frac{i}{2}, \frac{1}{2} - \frac{i}{2}$

Step 2.5.10

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

$\sin (x) = \frac{1}{2} + \frac{i}{2}$

$\sin (x) = \frac{1}{2} - \frac{i}{2}$

Step 2.5.11

Solve for $x$ in $\sin (x) = \frac{1}{2} + \frac{i}{2}$ .

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Step 2.5.11.1

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

$x = \arcsin (\frac{1}{2} + \frac{i}{2})$

Step 2.5.11.2

The inverse sine of $\arcsin (\frac{1}{2} + \frac{i}{2})$ is undefined.

Undefined

Step 2.5.12

Solve for $x$ in $\sin (x) = \frac{1}{2} - \frac{i}{2}$ .

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Step 2.5.12.1

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

$x = \arcsin (\frac{1}{2} - \frac{i}{2})$

Step 2.5.12.2

The inverse sine of $\arcsin (\frac{1}{2} - \frac{i}{2})$ is undefined.

Undefined

Step 2.5.13

List all of the solutions.

No solution