Trigonometry Examples

$\sin (2 x) = \frac{5 π}{4}$

Step 1

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

$2 x = \arcsin (\frac{5 π}{4})$

Step 2

Divide each term in $2 x = \arcsin (\frac{5 π}{4})$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x = \arcsin (\frac{5 π}{4})$ by $2$ .

$\frac{2 x}{2} = \frac{\arcsin (\frac{5 π}{4})}{2}$

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.

$\frac{2 x}{2} = \frac{\arcsin (\frac{5 π}{4})}{2}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (\frac{5 π}{4})}{2}$

Step 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.

$2 x = π - \arcsin (\frac{5 π}{4})$

Step 4

Divide each term in $2 x = π - \arcsin (\frac{5 π}{4})$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = π - \arcsin (\frac{5 π}{4})$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2} + \frac{- \arcsin (\frac{5 π}{4})}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2} + \frac{- \arcsin (\frac{5 π}{4})}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2} + \frac{- \arcsin (\frac{5 π}{4})}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Move the negative in front of the fraction.

$x = \frac{π}{2} - \frac{\arcsin (\frac{5 π}{4})}{2}$

Step 5

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

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Step 5.1

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

$\frac{2 π}{| b |}$

Step 5.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 5.3

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

$\frac{2 π}{2}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 5.4.2

Divide $π$ by $1$ .

$π$

Step 6

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

$x = \frac{\arcsin (\frac{5 π}{4})}{2} + π n, \frac{π}{2} - \frac{\arcsin (\frac{5 π}{4})}{2} + π n$ , for any integer $n$