Trigonometry Examples

$\sin (y) = \frac{2}{\sqrt{5}}$

Step 1

Simplify $\frac{2}{\sqrt{5}}$ .

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Step 1.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (y) = \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (y) = \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 1.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y) = \frac{2 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 1.2.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sin (y) = \frac{2 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 1.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (y) = \frac{2 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 1.2.5

Add $1$ and $1$ .

$\sin (y) = \frac{2 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 1.2.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sin (y) = \frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 1.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (y) = \frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 1.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sin (y) = \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$\sin (y) = \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.2.6.4.2

Rewrite the expression.

$\sin (y) = \frac{2 \sqrt{5}}{5^{1}}$

Step 1.2.6.5

Evaluate the exponent.

$\sin (y) = \frac{2 \sqrt{5}}{5}$

Step 2

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

$y = \arcsin (\frac{2 \sqrt{5}}{5})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (\frac{2 \sqrt{5}}{5})$ .

$y = 1.10714871$

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.

$y = (3.14159265) - 1.10714871$

Step 5

Solve for $y$ .

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Step 5.1

Remove parentheses.

$y = 3.14159265 - 1.10714871$

Step 5.2

Remove parentheses.

$y = (3.14159265) - 1.10714871$

Step 5.3

Subtract $1.10714871$ from $3.14159265$ .

$y = 2.03444393$

Step 6

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

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

$y = 1.10714871 + 2 π n, 2.03444393 + 2 π n$ , for any integer $n$