Trigonometry Examples

$\sin (2 x) = \frac{3}{4}$

Step 1

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

$2 x = \arcsin (\frac{3}{4})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arcsin (\frac{3}{4})$ .

$2 x = 0.84806207$

Step 3

Divide each term in $2 x = 0.84806207$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 x = 0.84806207$ by $2$ .

$\frac{2 x}{2} = \frac{0.84806207}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0.84806207}{2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.84806207}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0.84806207$ by $2$ .

$x = 0.42403103$

Step 4

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.

$2 x = (3.14159265) - 0.84806207$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $0.84806207$ from $3.14159265$ .

$2 x = 2.29353057$

Step 5.2

Divide each term in $2 x = 2.29353057$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 x = 2.29353057$ by $2$ .

$\frac{2 x}{2} = \frac{2.29353057}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2.29353057}{2}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.29353057}{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $2.29353057$ by $2$ .

$x = 1.14676528$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 2 |}$

Step 6.3

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

$\frac{2 π}{2}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 6.4.2

Divide $π$ by $1$ .

$π$

Step 7

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

$x = 0.42403103 + π n, 1.14676528 + π n$ , for any integer $n$