Trigonometry Examples

$\sin^{2} (x) - 2 \sin (x) + 3 = 0$

Step 1

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

${(u)}^{2} - 2 u + 3 = 0$

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 $a = 1$ , $b = - 2$ , and $c = 3$ into the quadratic formula and solve for $u$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$u = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{2 \pm \sqrt{4 - 4 \cdot 3}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $3$ .

$u = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 1}$

Step 4.1.3

Subtract $12$ from $4$ .

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

Step 4.1.4

Rewrite $- 8$ as $- 1 (8)$ .

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

Step 4.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

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

Step 4.1.6

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

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

Step 4.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.1.7.1

Factor $4$ out of $8$ .

$u = \frac{2 \pm i \cdot \sqrt{4 (2)}}{2 \cdot 1}$

Step 4.1.7.2

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

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

Step 4.1.8

Pull terms out from under the radical.

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

Step 4.1.9

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

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

Step 4.2

Multiply $2$ by $1$ .

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

Step 4.3

Simplify $\frac{2 \pm 2 i \sqrt{2}}{2}$ .

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

Step 5

The final answer is the combination of both solutions.

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

Step 6

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

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

Step 7

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

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

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

Step 8

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

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Step 8.1

Convert the right side of the equation to its decimal equivalent.

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

Step 8.2

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

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

Step 8.3

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

Undefined

Step 9

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

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Step 9.1

Convert the right side of the equation to its decimal equivalent.

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

Step 9.2

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

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

Step 9.3

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

Undefined

Step 10

List all of the solutions.

No solution