Trigonometry Examples

$2 \sin^{2} (x) + 1 = 4 \sin (x)$

Step 1

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

$2 {(u)}^{2} + 1 = 4 (u)$

Step 2

Subtract $4 u$ from both sides of the equation.

$2 u^{2} + 1 - 4 u = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 2$ , $b = - 4$ , and $c = 1$ into the quadratic formula and solve for $u$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 4$ to the power of $2$ .

$u = \frac{4 \pm \sqrt{16 - 4 \cdot 2 \cdot 1}}{2 \cdot 2}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$u = \frac{4 \pm \sqrt{16 - 8 \cdot 1}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $1$ .

$u = \frac{4 \pm \sqrt{16 - 8}}{2 \cdot 2}$

Step 5.1.3

Subtract $8$ from $16$ .

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

Step 5.1.4

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

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Step 5.1.4.1

Factor $4$ out of $8$ .

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

Step 5.1.4.2

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

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

Step 5.1.5

Pull terms out from under the radical.

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

Step 5.2

Multiply $2$ by $2$ .

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

Step 5.3

Simplify $\frac{4 \pm 2 \sqrt{2}}{4}$ .

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

Step 6

The final answer is the combination of both solutions.

$u = \frac{2 + \sqrt{2}}{2}, \frac{2 - \sqrt{2}}{2}$

Step 7

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

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

Step 8

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

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

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

Step 9

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

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Step 9.1

The range of sine is $- 1 \leq y \leq 1$ . Since $\frac{2 + \sqrt{2}}{2}$ does not fall in this range, there is no solution.

No solution

Step 10

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

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Step 10.1

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

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

Step 10.2

Simplify the right side.

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Step 10.2.1

Evaluate $\arcsin (\frac{2 - \sqrt{2}}{2})$ .

$x = 0.29725136$

Step 10.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) - 0.29725136$

Step 10.4

Solve for $x$ .

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Step 10.4.1

Remove parentheses.

$x = 3.14159265 - 0.29725136$

Step 10.4.2

Remove parentheses.

$x = (3.14159265) - 0.29725136$

Step 10.4.3

Subtract $0.29725136$ from $3.14159265$ .

$x = 2.84434129$

Step 10.5

Find the period of $\sin (x)$ .

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Step 10.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 10.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 10.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 10.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 10.6

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.29725136 + 2 π n, 2.84434129 + 2 π n$ , for any integer $n$

Step 11

List all of the solutions.

$x = 0.29725136 + 2 π n, 2.84434129 + 2 π n$ , for any integer $n$