Trigonometry Examples

$\frac{1}{32} \cdot ({(100)}^{2} \sin (2 x)) = 300$

Step 1

Multiply both sides of the equation by $\frac{1}{\frac{1}{32} \cdot 100^{2}}$ .

$\frac{1}{\frac{1}{32} \cdot 100^{2}} (\frac{1}{32} \cdot (100^{2} \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{1}{\frac{1}{32} \cdot 100^{2}} (\frac{1}{32} \cdot (100^{2} \sin (2 x)))$ .

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Step 2.1.1.1

Combine $\frac{1}{32}$ and $100^{2}$ .

$\frac{1}{\frac{100^{2}}{32}} (\frac{1}{32} \cdot (100^{2} \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.2

Raise $100$ to the power of $2$ .

$\frac{1}{\frac{10000}{32}} (\frac{1}{32} \cdot (100^{2} \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.3

Cancel the common factor of $10000$ and $32$ .

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Step 2.1.1.3.1

Factor $16$ out of $10000$ .

$\frac{1}{\frac{16 (625)}{32}} (\frac{1}{32} \cdot (100^{2} \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.3.2

Cancel the common factors.

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Step 2.1.1.3.2.1

Factor $16$ out of $32$ .

$\frac{1}{\frac{16 \cdot 625}{16 \cdot 2}} (\frac{1}{32} \cdot (100^{2} \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.3.2.2

Cancel the common factor.

$\frac{1}{\frac{16 \cdot 625}{16 \cdot 2}} (\frac{1}{32} \cdot (100^{2} \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.3.2.3

Rewrite the expression.

$\frac{1}{\frac{625}{2}} (\frac{1}{32} \cdot (100^{2} \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.4

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{2}{625}) (\frac{1}{32} \cdot (100^{2} \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.5

Simplify the expression.

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Step 2.1.1.5.1

Multiply $\frac{2}{625}$ by $1$ .

$\frac{2}{625} (\frac{1}{32} \cdot (100^{2} \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.5.2

Raise $100$ to the power of $2$ .

$\frac{2}{625} (\frac{1}{32} \cdot (10000 \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.6

Cancel the common factor of $625$ .

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Step 2.1.1.6.1

Factor $625$ out of $\frac{1}{32} \cdot (10000 \sin (2 x))$ .

$\frac{2}{625} (625 (\frac{1}{32} \cdot (16 \sin (2 x)))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.6.2

Cancel the common factor.

$\frac{2}{625} (625 (\frac{1}{32} \cdot (16 \sin (2 x)))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.6.3

Rewrite the expression.

$2 (\frac{1}{32} \cdot (16 \sin (2 x))) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.7

Combine $16$ and $\frac{1}{32}$ .

$2 (\frac{16}{32} \cdot \sin (2 x)) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.8

Combine $\frac{16}{32}$ and $\sin (2 x)$ .

$2 \frac{16 \sin (2 x)}{32} = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.9

Combine $2$ and $\frac{16 \sin (2 x)}{32}$ .

$\frac{2 (16 \sin (2 x))}{32} = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.10

Multiply $16$ by $2$ .

$\frac{32 \sin (2 x)}{32} = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.11

Cancel the common factor of $32$ .

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Step 2.1.1.11.1

Cancel the common factor.

$\frac{32 \sin (2 x)}{32} = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.1.1.11.2

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

$\sin (2 x) = \frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{1}{\frac{1}{32} \cdot 100^{2}} \cdot 300$ .

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Step 2.2.1.1

Combine $\frac{1}{32}$ and $100^{2}$ .

$\sin (2 x) = \frac{1}{\frac{100^{2}}{32}} \cdot 300$

Step 2.2.1.2

Raise $100$ to the power of $2$ .

$\sin (2 x) = \frac{1}{\frac{10000}{32}} \cdot 300$

Step 2.2.1.3

Cancel the common factor of $10000$ and $32$ .

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Step 2.2.1.3.1

Factor $16$ out of $10000$ .

$\sin (2 x) = \frac{1}{\frac{16 (625)}{32}} \cdot 300$

Step 2.2.1.3.2

Cancel the common factors.

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Step 2.2.1.3.2.1

Factor $16$ out of $32$ .

$\sin (2 x) = \frac{1}{\frac{16 \cdot 625}{16 \cdot 2}} \cdot 300$

Step 2.2.1.3.2.2

Cancel the common factor.

$\sin (2 x) = \frac{1}{\frac{16 \cdot 625}{16 \cdot 2}} \cdot 300$

Step 2.2.1.3.2.3

Rewrite the expression.

$\sin (2 x) = \frac{1}{\frac{625}{2}} \cdot 300$

Step 2.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$\sin (2 x) = 1 (\frac{2}{625}) \cdot 300$

Step 2.2.1.5

Multiply $\frac{2}{625}$ by $1$ .

$\sin (2 x) = \frac{2}{625} \cdot 300$

Step 2.2.1.6

Cancel the common factor of $25$ .

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Step 2.2.1.6.1

Factor $25$ out of $625$ .

$\sin (2 x) = \frac{2}{25 (25)} \cdot 300$

Step 2.2.1.6.2

Factor $25$ out of $300$ .

$\sin (2 x) = \frac{2}{25 \cdot 25} \cdot (25 \cdot 12)$

Step 2.2.1.6.3

Cancel the common factor.

$\sin (2 x) = \frac{2}{25 \cdot 25} \cdot (25 \cdot 12)$

Step 2.2.1.6.4

Rewrite the expression.

$\sin (2 x) = \frac{2}{25} \cdot 12$

Step 2.2.1.7

Combine $\frac{2}{25}$ and $12$ .

$\sin (2 x) = \frac{2 \cdot 12}{25}$

Step 2.2.1.8

Multiply $2$ by $12$ .

$\sin (2 x) = \frac{24}{25}$

Step 3

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

$2 x = \arcsin (\frac{24}{25})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arcsin (\frac{24}{25})$ .

$2 x = 1.28700221$

Step 5

Divide each term in $2 x = 1.28700221$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = 1.28700221$ by $2$ .

$\frac{2 x}{2} = \frac{1.28700221}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1.28700221}{2}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.28700221}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $1.28700221$ by $2$ .

$x = 0.6435011$

Step 6

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.

$2 x = (3.14159265) - 1.28700221$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $1.28700221$ from $3.14159265$ .

$2 x = 1.85459043$

Step 7.2

Divide each term in $2 x = 1.85459043$ by $2$ and simplify.

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Step 7.2.1

Divide each term in $2 x = 1.85459043$ by $2$ .

$\frac{2 x}{2} = \frac{1.85459043}{2}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1.85459043}{2}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1.85459043}{2}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $1.85459043$ by $2$ .

$x = 0.92729521$

Step 8

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

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

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

$\frac{2 π}{| 2 |}$

Step 8.3

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

$\frac{2 π}{2}$

Step 8.4

Cancel the common factor of $2$ .

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Step 8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 8.4.2

Divide $π$ by $1$ .

$π$

Step 9

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

$x = 0.6435011 + π n, 0.92729521 + π n$ , for any integer $n$