Trigonometry Examples

$\sin (x) = - \frac{\sqrt{5}}{5}$ , $\frac{3 π}{2} < x < 2 π$

Step 1

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

$x = \arcsin (- \frac{\sqrt{5}}{5})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arcsin (- \frac{\sqrt{5}}{5})$ .

$x = - 0.4636476$

Step 3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) + 0.4636476 + 3.14159265$

Step 4

Simplify the expression to find the second solution.

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Step 4.1

Subtract $2 π$ from $2 (3.14159265) + 0.4636476 + 3.14159265$ .

$x = 2 (3.14159265) + 0.4636476 + 3.14159265 - 2 π$

Step 4.2

The resulting angle of $3.60524026$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.4636476 + 3.14159265$ .

$x = 3.60524026$

Step 5

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

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6

Add $2 π$ to every negative angle to get positive angles.

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Step 6.1

Add $2 π$ to $- 0.4636476$ to find the positive angle.

$- 0.4636476 + 2 π$

Step 6.2

Subtract $0.4636476$ from $2 π$ .

$5.81953769$

Step 6.3

List the new angles.

$x = 5.81953769$

Step 7

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

$x = 3.60524026 + 2 π n, 5.81953769 + 2 π n$ , for any integer $n$

Step 8

Plug in $0$ for $n$ and simplify to see if the solution is contained in $(\frac{3 π}{2}, 2 π)$ .

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Step 8.1

Plug in $0$ for $n$ .

$5.81953769 + 2 π (0)$

Step 8.2

Simplify.

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Step 8.2.1

Multiply $2 π (0)$ .

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Step 8.2.1.1

Multiply $0$ by $2$ .

$5.81953769 + 0 π$

Step 8.2.1.2

Multiply $0$ by $π$ .

$5.81953769 + 0$

Step 8.2.2

Add $5.81953769$ and $0$ .

$5.81953769$

Step 8.3

The interval $(\frac{3 π}{2}, 2 π)$ contains $5.81953769$ .

$x = 5.81953769$