Trigonometry Examples

$2 \sin (4 x) + 6 = 5$

Step 1

Move all terms not containing $\sin (4 x)$ to the right side of the equation.

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Step 1.1

Subtract $6$ from both sides of the equation.

$2 \sin (4 x) = 5 - 6$

Step 1.2

Subtract $6$ from $5$ .

$2 \sin (4 x) = - 1$

Step 2

Divide each term in $2 \sin (4 x) = - 1$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 \sin (4 x) = - 1$ by $2$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $\sin (4 x)$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

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

$4 x = \arcsin (- \frac{1}{2})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arcsin (- \frac{1}{2})$ is $- 30$ .

$4 x = - 30$

Step 5

Divide each term in $4 x = - 30$ by $4$ and simplify.

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Step 5.1

Divide each term in $4 x = - 30$ by $4$ .

$\frac{4 x}{4} = \frac{- 30}{4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 30}{4}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 30}{4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $- 30$ and $4$ .

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Step 5.3.1.1

Factor $2$ out of $- 30$ .

$x = \frac{2 (- 15)}{4}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot - 15}{2 \cdot 2}$

Step 5.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 15}{2 \cdot 2}$

Step 5.3.1.2.3

Rewrite the expression.

$x = \frac{- 15}{2}$

Step 5.3.2

Move the negative in front of the fraction.

$x = - \frac{15}{2}$

Step 6

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $360$ , to find a reference angle. Next, add this reference angle to $180$ to find the solution in the third quadrant.

$4 x = 360 + 30 + 180$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

Subtract $360 °$ from $360 + 30 + 180 °$ .

$4 x = 360 + 30 + 180 ° - 360 °$

Step 7.2

The resulting angle of $210 °$ is positive, less than $360 °$ , and coterminal with $360 + 30 + 180$ .

$4 x = 210 °$

Step 7.3

Divide each term in $4 x = 210 °$ by $4$ and simplify.

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Step 7.3.1

Divide each term in $4 x = 210 °$ by $4$ .

$\frac{4 x}{4} = \frac{210 °}{4}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $4$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{210 °}{4}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{210 °}{4}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Cancel the common factor of $210 °$ and $4$ .

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Step 7.3.3.1.1

Factor $2$ out of $210 °$ .

$x = \frac{2 (105)}{4}$

Step 7.3.3.1.2

Cancel the common factors.

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Step 7.3.3.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 105}{2 \cdot 2}$

Step 7.3.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 105}{2 \cdot 2}$

Step 7.3.3.1.2.3

Rewrite the expression.

$x = \frac{105}{2}$

Step 8

Find the period of $\sin (4 x)$ .

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Step 8.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 8.2

Replace $b$ with $4$ in the formula for period.

$\frac{360}{| 4 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{360}{4}$

Step 8.4

Divide $360$ by $4$ .

$90$

Step 9

Add $90$ to every negative angle to get positive angles.

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Step 9.1

Add $90$ to $- \frac{15}{2}$ to find the positive angle.

$- \frac{15}{2} + 90$

Step 9.2

To write $90$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$90 \cdot \frac{2}{2} - \frac{15}{2}$

Step 9.3

Combine $90$ and $\frac{2}{2}$ .

$\frac{90 \cdot 2}{2} - \frac{15}{2}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{90 \cdot 2 - 15}{2}$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Multiply $90$ by $2$ .

$\frac{180 - 15}{2}$

Step 9.5.2

Subtract $15$ from $180$ .

$\frac{165}{2}$

Step 9.6

List the new angles.

$x = \frac{165}{2}$

Step 10

The period of the $\sin (4 x)$ function is $90$ so values will repeat every $90$ degrees in both directions.

$x = \frac{105}{2} + 90 n, \frac{165}{2} + 90 n$ , for any integer $n$