Trigonometry Examples

$560 = \frac{82^{2} \sin (2 x)}{9.8}$

Step 1

Rewrite the equation as $\frac{82^{2} \sin (2 x)}{9.8} = 560$ .

$\frac{82^{2} \sin (2 x)}{9.8} = 560$

Step 2

Multiply both sides of the equation by $\frac{9.8}{82^{2}}$ .

$\frac{9.8}{82^{2}} \cdot \frac{82^{2} \sin (2 x)}{9.8} = \frac{9.8}{82^{2}} \cdot 560$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{9.8}{82^{2}} \cdot \frac{82^{2} \sin (2 x)}{9.8}$ .

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Step 3.1.1.1

Combine.

$\frac{9.8 (82^{2} \sin (2 x))}{82^{2} \cdot 9.8} = \frac{9.8}{82^{2}} \cdot 560$

Step 3.1.1.2

Cancel the common factor of $9.8$ .

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Step 3.1.1.2.1

Cancel the common factor.

$\frac{9.8 (82^{2} \sin (2 x))}{82^{2} \cdot 9.8} = \frac{9.8}{82^{2}} \cdot 560$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{82^{2} \sin (2 x)}{82^{2}} = \frac{9.8}{82^{2}} \cdot 560$

Step 3.1.1.3

Cancel the common factor of $82^{2}$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{82^{2} \sin (2 x)}{82^{2}} = \frac{9.8}{82^{2}} \cdot 560$

Step 3.1.1.3.2

Divide $\sin (2 x)$ by $1$ .

$\sin (2 x) = \frac{9.8}{82^{2}} \cdot 560$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{9.8}{82^{2}} \cdot 560$ .

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Step 3.2.1.1

Raise $82$ to the power of $2$ .

$\sin (2 x) = \frac{9.8}{6724} \cdot 560$

Step 3.2.1.2

Cancel the common factor of $4$ .

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Step 3.2.1.2.1

Factor $4$ out of $6724$ .

$\sin (2 x) = \frac{9.8}{4 (1681)} \cdot 560$

Step 3.2.1.2.2

Factor $4$ out of $560$ .

$\sin (2 x) = \frac{9.8}{4 \cdot 1681} \cdot (4 \cdot 140)$

Step 3.2.1.2.3

Cancel the common factor.

$\sin (2 x) = \frac{9.8}{4 \cdot 1681} \cdot (4 \cdot 140)$

Step 3.2.1.2.4

Rewrite the expression.

$\sin (2 x) = \frac{9.8}{1681} \cdot 140$

Step 3.2.1.3

Combine $\frac{9.8}{1681}$ and $140$ .

$\sin (2 x) = \frac{9.8 \cdot 140}{1681}$

Step 3.2.1.4

Multiply $9.8$ by $140$ .

$\sin (2 x) = \frac{1372}{1681}$

Step 3.2.1.5

Cancel the common factor of $1372$ and $1681$ .

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Step 3.2.1.5.1

Rewrite $1372$ as $1 (1372)$ .

$\sin (2 x) = \frac{1 (1372)}{1681}$

Step 3.2.1.5.2

Cancel the common factors.

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Step 3.2.1.5.2.1

Rewrite $1681$ as $1 (1681)$ .

$\sin (2 x) = \frac{1 \cdot 1372}{1 \cdot 1681}$

Step 3.2.1.5.2.2

Cancel the common factor.

$\sin (2 x) = \frac{1 \cdot 1372}{1 \cdot 1681}$

Step 3.2.1.5.2.3

Rewrite the expression.

$\sin (2 x) = \frac{1372}{1681}$

Step 4

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

$2 x = \arcsin (\frac{1372}{1681})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (\frac{1372}{1681})$ .

$2 x = 0.95476995$

Step 6

Divide each term in $2 x = 0.95476995$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 x = 0.95476995$ by $2$ .

$\frac{2 x}{2} = \frac{0.95476995}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0.95476995}{2}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.95476995}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0.95476995$ by $2$ .

$x = 0.47738497$

Step 7

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.

$2 x = (3.14159265) - 0.95476995$

Step 8

Solve for $x$ .

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Step 8.1

Subtract $0.95476995$ from $3.14159265$ .

$2 x = 2.18682269$

Step 8.2

Divide each term in $2 x = 2.18682269$ by $2$ and simplify.

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Step 8.2.1

Divide each term in $2 x = 2.18682269$ by $2$ .

$\frac{2 x}{2} = \frac{2.18682269}{2}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $2$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2.18682269}{2}$

Step 8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.18682269}{2}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Divide $2.18682269$ by $2$ .

$x = 1.09341134$

Step 9

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

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

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

$\frac{2 π}{| 2 |}$

Step 9.3

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

$\frac{2 π}{2}$

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 9.4.2

Divide $π$ by $1$ .

$π$

Step 10

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

$x = 0.47738497 + π n, 1.09341134 + π n$ , for any integer $n$