Finite Math Examples

$\sin (2 a) = \frac{130}{280^{2}}$

Step 1

Simplify $\frac{130}{280^{2}}$ .

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Step 1.1

Raise $280$ to the power of $2$ .

$\sin (2 a) = \frac{130}{78400}$

Step 1.2

Cancel the common factor of $130$ and $78400$ .

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Step 1.2.1

Factor $10$ out of $130$ .

$\sin (2 a) = \frac{10 (13)}{78400}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $10$ out of $78400$ .

$\sin (2 a) = \frac{10 \cdot 13}{10 \cdot 7840}$

Step 1.2.2.2

Cancel the common factor.

$\sin (2 a) = \frac{10 \cdot 13}{10 \cdot 7840}$

Step 1.2.2.3

Rewrite the expression.

$\sin (2 a) = \frac{13}{7840}$

Step 2

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

$2 a = \arcsin (\frac{13}{7840})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arcsin (\frac{13}{7840})$ .

$2 a = 0.00165816$

Step 4

Divide each term in $2 a = 0.00165816$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 a = 0.00165816$ by $2$ .

$\frac{2 a}{2} = \frac{0.00165816}{2}$

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.

$\frac{2 a}{2} = \frac{0.00165816}{2}$

Step 4.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.00165816}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0.00165816$ by $2$ .

$a = 0.00082908$

Step 5

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.

$2 a = (3.14159265) - 0.00165816$

Step 6

Solve for $a$ .

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Step 6.1

Subtract $0.00165816$ from $3.14159265$ .

$2 a = 3.13993448$

Step 6.2

Divide each term in $2 a = 3.13993448$ by $2$ and simplify.

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Step 6.2.1

Divide each term in $2 a = 3.13993448$ by $2$ .

$\frac{2 a}{2} = \frac{3.13993448}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{3.13993448}{2}$

Step 6.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{3.13993448}{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $3.13993448$ by $2$ .

$a = 1.56996724$

Step 7

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

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 2 |}$

Step 7.3

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

$\frac{2 π}{2}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 7.4.2

Divide $π$ by $1$ .

$π$

Step 8

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

$a = 0.00082908 + π n, 1.56996724 + π n$ , for any integer $n$