Trigonometry Examples

${(7 \sin (x))}^{2} - \frac{4 \sin (2 x)}{\cos (x)} = - 1$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the product rule to $7 \sin (x)$ .

$7^{2} \sin^{2} (x) - \frac{4 \sin (2 x)}{\cos (x)} = - 1$

Step 1.1.2

Raise $7$ to the power of $2$ .

$49 \sin^{2} (x) - \frac{4 \sin (2 x)}{\cos (x)} = - 1$

Step 1.1.3

Simplify the numerator.

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Step 1.1.3.1

Apply the sine double-angle identity.

$49 \sin^{2} (x) - \frac{4 \cdot 2 \sin (x) \cos (x)}{\cos (x)} = - 1$

Step 1.1.3.2

Multiply $4$ by $2$ .

$49 \sin^{2} (x) - \frac{8 \sin (x) \cos (x)}{\cos (x)} = - 1$

Step 1.1.4

Cancel the common factor of $\cos (x)$ .

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Step 1.1.4.1

Cancel the common factor.

$49 \sin^{2} (x) - \frac{8 \sin (x) \cos (x)}{\cos (x)} = - 1$

Step 1.1.4.2

Divide $8 \sin (x)$ by $1$ .

$49 \sin^{2} (x) - (8 \sin (x)) = - 1$

Step 1.1.5

Multiply $8$ by $- 1$ .

$49 \sin^{2} (x) - 8 \sin (x) = - 1$

Step 2

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

$49 {(u)}^{2} - 8 u = - 1$

Step 3

Add $1$ to both sides of the equation.

$49 u^{2} - 8 u + 1 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 49$ , $b = - 8$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (49 \cdot 1)}}{2 \cdot 49}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 8$ to the power of $2$ .

$u = \frac{8 \pm \sqrt{64 - 4 \cdot 49 \cdot 1}}{2 \cdot 49}$

Step 6.1.2

Multiply $- 4 \cdot 49 \cdot 1$ .

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Step 6.1.2.1

Multiply $- 4$ by $49$ .

$u = \frac{8 \pm \sqrt{64 - 196 \cdot 1}}{2 \cdot 49}$

Step 6.1.2.2

Multiply $- 196$ by $1$ .

$u = \frac{8 \pm \sqrt{64 - 196}}{2 \cdot 49}$

Step 6.1.3

Subtract $196$ from $64$ .

$u = \frac{8 \pm \sqrt{- 132}}{2 \cdot 49}$

Step 6.1.4

Rewrite $- 132$ as $- 1 (132)$ .

$u = \frac{8 \pm \sqrt{- 1 \cdot 132}}{2 \cdot 49}$

Step 6.1.5

Rewrite $\sqrt{- 1 (132)}$ as $\sqrt{- 1} \cdot \sqrt{132}$ .

$u = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{132}}{2 \cdot 49}$

Step 6.1.6

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

$u = \frac{8 \pm i \cdot \sqrt{132}}{2 \cdot 49}$

Step 6.1.7

Rewrite $132$ as $2^{2} \cdot 33$ .

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Step 6.1.7.1

Factor $4$ out of $132$ .

$u = \frac{8 \pm i \cdot \sqrt{4 (33)}}{2 \cdot 49}$

Step 6.1.7.2

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

$u = \frac{8 \pm i \cdot \sqrt{2^{2} \cdot 33}}{2 \cdot 49}$

Step 6.1.8

Pull terms out from under the radical.

$u = \frac{8 \pm i \cdot (2 \sqrt{33})}{2 \cdot 49}$

Step 6.1.9

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

$u = \frac{8 \pm 2 i \sqrt{33}}{2 \cdot 49}$

Step 6.2

Multiply $2$ by $49$ .

$u = \frac{8 \pm 2 i \sqrt{33}}{98}$

Step 6.3

Simplify $\frac{8 \pm 2 i \sqrt{33}}{98}$ .

$u = \frac{4 \pm i \sqrt{33}}{49}$

Step 7

The final answer is the combination of both solutions.

$u = \frac{4 + i \sqrt{33}}{49}, \frac{4 - i \sqrt{33}}{49}$

Step 8

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

$\sin (x) = \frac{4 + i \sqrt{33}}{49}, \frac{4 - i \sqrt{33}}{49}$

Step 9

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

$\sin (x) = \frac{4 + i \sqrt{33}}{49}$

$\sin (x) = \frac{4 - i \sqrt{33}}{49}$

Step 10

Solve for $x$ in $\sin (x) = \frac{4 + i \sqrt{33}}{49}$ .

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Step 10.1

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

$x = \arcsin (\frac{4 + i \sqrt{33}}{49})$

Step 10.2

The inverse sine of $\arcsin (\frac{4 + i \sqrt{33}}{49})$ is undefined.

Undefined

Step 11

Solve for $x$ in $\sin (x) = \frac{4 - i \sqrt{33}}{49}$ .

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Step 11.1

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

$x = \arcsin (\frac{4 - i \sqrt{33}}{49})$

Step 11.2

The inverse sine of $\arcsin (\frac{4 - i \sqrt{33}}{49})$ is undefined.

Undefined

Step 12

List all of the solutions.

No solution