Trigonometry Examples

$3 - \sin (x) = \cos (2 x)$

Step 1

Use the double-angle identity to transform

$\cos (2 x)$ to

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

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

Step 2

Subtract

$3$ from both sides of the equation.

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

Step 3

Add

$2 \sin^{2} (x)$ to both sides of the equation.

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

Step 4

Simplify the right side.

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Step 4.1

Subtract

$3$ from

$1$ .

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

Step 5

Solve the equation for

$x$ .

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Step 5.1

Substitute

$u$ for

$\sin (x)$ .

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

Step 5.2

Add

$2$ to both sides of the equation.

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

Step 5.3

Use the quadratic formula to find the solutions.

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

Step 5.4

Substitute the values

$a = 2$ ,

$b = - 1$ , and

$c = 2$ into the quadratic formula and solve for

$u$ .

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

Step 5.5

Simplify.

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

Raise

$- 1$ to the power of

$2$ .

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

Step 5.5.1.2

Multiply

$- 4 \cdot 2 \cdot 2$ .

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Step 5.5.1.2.1

Multiply

$- 4$ by

$2$ .

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

Step 5.5.1.2.2

Multiply

$- 8$ by

$2$ .

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

Step 5.5.1.3

Subtract

$16$ from

$1$ .

$u = \frac{1 \pm \sqrt{- 15}}{2 \cdot 2}$

Step 5.5.1.4

Rewrite

$- 15$ as

$- 1 (15)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

Step 5.5.1.5

Rewrite

$\sqrt{- 1 (15)}$ as

$\sqrt{- 1} \cdot \sqrt{15}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

Step 5.5.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$u = \frac{1 \pm i \sqrt{15}}{2 \cdot 2}$

Step 5.5.2

Multiply

$2$ by

$2$ .

$u = \frac{1 \pm i \sqrt{15}}{4}$

Step 5.6

The final answer is the combination of both solutions.

$u = \frac{1 + i \sqrt{15}}{4}, \frac{1 - i \sqrt{15}}{4}$

Step 5.7

Substitute

$\sin (x)$ for

$u$ .

$\sin (x) = \frac{1 + i \sqrt{15}}{4}, \frac{1 - i \sqrt{15}}{4}$

Step 5.8

Set up each of the solutions to solve for

$x$ .

$\sin (x) = \frac{1 + i \sqrt{15}}{4}$

$\sin (x) = \frac{1 - i \sqrt{15}}{4}$

Step 5.9

Solve for

$x$ in

$\sin (x) = \frac{1 + i \sqrt{15}}{4}$ .

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Step 5.9.1

Take the inverse sine of both sides of the equation to extract

$x$ from inside the sine.

$x = \arcsin (\frac{1 + i \sqrt{15}}{4})$

Step 5.9.2

The inverse sine of

$\arcsin (\frac{1 + i \sqrt{15}}{4})$ is undefined.

Undefined

Step 5.10

Solve for

$x$ in

$\sin (x) = \frac{1 - i \sqrt{15}}{4}$ .

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Step 5.10.1

Take the inverse sine of both sides of the equation to extract

$x$ from inside the sine.

$x = \arcsin (\frac{1 - i \sqrt{15}}{4})$

Step 5.10.2

The inverse sine of

$\arcsin (\frac{1 - i \sqrt{15}}{4})$ is undefined.

Undefined

Step 5.11

List all of the solutions.

No solution