Trigonometry Examples

$2 \sin^{2} (x) - \sin (x) + 8 = 0$

Step 1

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

$2 {(u)}^{2} - (u) + 8 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 2$ , $b = - 1$ , and $c = 8$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (2 \cdot 8)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 1$ to the power of $2$ .

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

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 8$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

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

Step 4.1.2.2

Multiply $- 8$ by $8$ .

$u = \frac{1 \pm \sqrt{1 - 64}}{2 \cdot 2}$

Step 4.1.3

Subtract $64$ from $1$ .

$u = \frac{1 \pm \sqrt{- 63}}{2 \cdot 2}$

Step 4.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$u = \frac{1 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 2}$

Step 4.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 2}$

Step 4.1.6

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

$u = \frac{1 \pm i \cdot \sqrt{63}}{2 \cdot 2}$

Step 4.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 4.1.7.1

Factor $9$ out of $63$ .

$u = \frac{1 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 2}$

Step 4.1.7.2

Rewrite $9$ as $3^{2}$ .

$u = \frac{1 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 2}$

Step 4.1.8

Pull terms out from under the radical.

$u = \frac{1 \pm i \cdot (3 \sqrt{7})}{2 \cdot 2}$

Step 4.1.9

Move $3$ to the left of $i$ .

$u = \frac{1 \pm 3 i \sqrt{7}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$u = \frac{1 \pm 3 i \sqrt{7}}{4}$

Step 5

The final answer is the combination of both solutions.

$u = \frac{1 + 3 i \sqrt{7}}{4}, \frac{1 - 3 i \sqrt{7}}{4}$

Step 6

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

$\sin (x) = \frac{1 + 3 i \sqrt{7}}{4}, \frac{1 - 3 i \sqrt{7}}{4}$

Step 7

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

$\sin (x) = \frac{1 + 3 i \sqrt{7}}{4}$

$\sin (x) = \frac{1 - 3 i \sqrt{7}}{4}$

Step 8

Solve for $x$ in $\sin (x) = \frac{1 + 3 i \sqrt{7}}{4}$ .

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Step 8.1

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

$x = \arcsin (\frac{1 + 3 i \sqrt{7}}{4})$

Step 8.2

The inverse sine of $\arcsin (\frac{1 + 3 i \sqrt{7}}{4})$ is undefined.

Undefined

Step 9

Solve for $x$ in $\sin (x) = \frac{1 - 3 i \sqrt{7}}{4}$ .

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Step 9.1

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

$x = \arcsin (\frac{1 - 3 i \sqrt{7}}{4})$

Step 9.2

The inverse sine of $\arcsin (\frac{1 - 3 i \sqrt{7}}{4})$ is undefined.

Undefined

Step 10

List all of the solutions.

No solution