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.

Tap for more steps...

Step 4.1

Subtract $3$ from $1$ .

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

Step 5

Solve the equation for $x$ .

Tap for more steps...

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.

Tap for more steps...

Step 5.5.1

Simplify the numerator.

Tap for more steps...

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 $- 4 \cdot 2 \cdot 2$ .

Tap for more steps...

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

Tap for more steps...

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

Tap for more steps...

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