Trigonometry Examples

$\sin (2 x + 20) = \cos (30)$

Step 1

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\sin (2 x + 20) = \frac{\sqrt{3}}{2}$

Step 2

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

$2 x + 20 = \arcsin (\frac{\sqrt{3}}{2})$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arcsin (\frac{\sqrt{3}}{2})$ is $60$ .

$2 x + 20 = 60$

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

Subtract $20$ from both sides of the equation.

$2 x = 60 - 20$

Step 4.2

Subtract $20$ from $60$ .

$2 x = 40$

Step 5

Divide each term in $2 x = 40$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = 40$ by $2$ .

$\frac{2 x}{2} = \frac{40}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{40}{2}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{40}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $40$ by $2$ .

$x = 20$

Step 6

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$2 x + 20 = 180 - 60$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $60$ from $180$ .

$2 x + 20 = 120$

Step 7.2

Move all terms not containing $x$ to the right side of the equation.

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Step 7.2.1

Subtract $20$ from both sides of the equation.

$2 x = 120 - 20$

Step 7.2.2

Subtract $20$ from $120$ .

$2 x = 100$

Step 7.3

Divide each term in $2 x = 100$ by $2$ and simplify.

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Step 7.3.1

Divide each term in $2 x = 100$ by $2$ .

$\frac{2 x}{2} = \frac{100}{2}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $2$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{100}{2}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{100}{2}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Divide $100$ by $2$ .

$x = 50$

Step 8

Find the period of $\sin (2 x + 20)$ .

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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 $2$ in the formula for period.

$\frac{360}{| 2 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{360}{2}$

Step 8.4

Divide $360$ by $2$ .

$180$

Step 9

The period of the $\sin (2 x + 20)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$x = 20 + 180 n, 50 + 180 n$ , for any integer $n$