Precalculus Examples

$\sin (2 x) = y$ , $\sin (x) = \frac{3}{\sqrt{13}}$

Step 1

Simplify $\frac{3}{\sqrt{13}}$ .

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Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sin (x) = \frac{3}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Combine and simplify the denominator.

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Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sin (x) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$\sin (x) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$\sin (x) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}$

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

$\sin (x) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (x) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{2}}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sin (x) = \frac{3 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

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

$\sin (x) = \frac{3 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

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

$\sin (x) = \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\sin (x) = \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

Rewrite the expression.

$\sin (x) = \frac{3 \sqrt{13}}{13^{1}}$

Evaluate the exponent.

$\sin (x) = \frac{3 \sqrt{13}}{13}$

Step 2

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

$x = \arcsin (\frac{3 \sqrt{13}}{13})$

Step 3

Simplify the right side.

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Evaluate $\arcsin (\frac{3 \sqrt{13}}{13})$ .

$x = 0.98279372$

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.

$x = (3.14159265) - 0.98279372$

Step 5

Solve for $x$ .

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Remove parentheses.

$x = 3.14159265 - 0.98279372$

Remove parentheses.

$x = (3.14159265) - 0.98279372$

Subtract $0.98279372$ from $3.14159265$ .

$x = 2.15879893$

Step 6

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

$x = 0.98279372 + 2 π n, 2.15879893 + 2 π n$ , for any integer $n$

Step 8

Reorder $0.98279372$ and $2 π n$ .

$x = 2 π n + 0.98279372$

$2.15879893 + 2 π n$

Step 9

Solve the equation for $n$ .

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Simplify each term.

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Apply the distributive property.

$(2 \cdot 2.15879893 + 2 (2 π n)) \cdot n + 0.98279372 = 0.98279372 + 2 (2.15879893 + 2 π n) \cdot n$

$2.15879893 + 2 π n$

Multiply $2$ by $2.15879893$ .

$(4.31759786 + 2 (2 π n)) \cdot n + 0.98279372 = 0.98279372 + 2 (2.15879893 + 2 π n) \cdot n$

$2.15879893 + 2 π n$

Multiply $2$ by $2$ .

$(4.31759786 + 4 (π n)) \cdot n + 0.98279372 = 0.98279372 + 2 (2.15879893 + 2 π n) \cdot n$

$2.15879893 + 2 π n$

Apply the distributive property.

$4.31759786 n + 4 π n \cdot n + 0.98279372 = 0.98279372 + 2 (2.15879893 + 2 π n) \cdot n$

$2.15879893 + 2 π n$

Multiply $n$ by $n$ by adding the exponents.

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Move $n$ .

$4.31759786 n + 4 π (n \cdot n) + 0.98279372 = 0.98279372 + 2 (2.15879893 + 2 π n) \cdot n$

$2.15879893 + 2 π n$

Multiply $n$ by $n$ .

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + 2 (2.15879893 + 2 π n) \cdot n$

$2.15879893 + 2 π n$

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + 2 (2.15879893 + 2 π n) \cdot n$

$2.15879893 + 2 π n$

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + 2 (2.15879893 + 2 π n) \cdot n$

$2.15879893 + 2 π n$

Simplify each term.

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Apply the distributive property.

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + (2 \cdot 2.15879893 + 2 (2 π n)) \cdot n$

$2.15879893 + 2 π n$

Multiply $2$ by $2.15879893$ .

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + (4.31759786 + 2 (2 π n)) \cdot n$

$2.15879893 + 2 π n$

Multiply $2$ by $2$ .

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + (4.31759786 + 4 (π n)) \cdot n$

$2.15879893 + 2 π n$

Apply the distributive property.

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + 4.31759786 n + 4 π n \cdot n$

$2.15879893 + 2 π n$

Multiply $n$ by $n$ by adding the exponents.

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Move $n$ .

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + 4.31759786 n + 4 π (n \cdot n)$

$2.15879893 + 2 π n$

Multiply $n$ by $n$ .

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + 4.31759786 n + 4 π n^{2}$

$2.15879893 + 2 π n$

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + 4.31759786 n + 4 π n^{2}$

$2.15879893 + 2 π n$

$4.31759786 n + 4 π n^{2} + 0.98279372 = 0.98279372 + 4.31759786 n + 4 π n^{2}$

$2.15879893 + 2 π n$

Move all terms containing $n$ to the left side of the equation.

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Subtract $4.31759786 n$ from both sides of the equation.

$4.31759786 n + 4 π n^{2} + 0.98279372 - 4.31759786 n = 0.98279372 + 4 π n^{2}$

$2.15879893 + 2 π n$

Subtract $4 π n^{2}$ from both sides of the equation.

$4.31759786 n + 4 π n^{2} + 0.98279372 - 4.31759786 n - 4 π n^{2} = 0.98279372$

$2.15879893 + 2 π n$

Combine the opposite terms in $4.31759786 n + 4 π n^{2} + 0.98279372 - 4.31759786 n - 4 π n^{2}$ .

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Subtract $4.31759786 n$ from $4.31759786 n$ .

$0 n + 4 π n^{2} + 0.98279372 - 4 π n^{2} = 0.98279372$

$2.15879893 + 2 π n$

Subtract $4 π n^{2}$ from $4 π n^{2}$ .

$0 n + 0 + 0.98279372 = 0.98279372$

$2.15879893 + 2 π n$

Add $0 n$ and $0$ .

$0 n + 0.98279372 = 0.98279372$

$2.15879893 + 2 π n$

$0 n + 0.98279372 = 0.98279372$

$2.15879893 + 2 π n$

Multiply $0$ by $n$ .

$0 + 0.98279372 = 0.98279372$

$2.15879893 + 2 π n$

Add $0$ and $0.98279372$ .

$0.98279372 = 0.98279372$

$2.15879893 + 2 π n$

$0.98279372 = 0.98279372$

$2.15879893 + 2 π n$

Since $0.98279372 = 0.98279372$ , the equation will always be true.

Always true

$2.15879893 + 2 π n$

Always true

$2.15879893 + 2 π n$

Step 10

The simplified system is the arbitrary solution of the original system of equations.

Always true

$2.15879893 + 2 (2.15879893 + 2 π n) \cdot n$