Trigonometry Examples

$y = 15.8 \sin (\frac{π}{6} x) + 40.8$

Step 1

Rewrite the equation as $15.8 \sin (\frac{π}{6} x) + 40.8 = y$ .

$15.8 \sin (\frac{π}{6} x) + 40.8 = y$

Step 2

Subtract $40.8$ from both sides of the equation.

$15.8 \sin (\frac{π}{6} x) = y - 40.8$

Step 3

Divide each term in $15.8 \sin (\frac{π}{6} x) = y - 40.8$ by $15.8$ and simplify.

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Step 3.1

Divide each term in $15.8 \sin (\frac{π}{6} x) = y - 40.8$ by $15.8$ .

$\frac{15.8 \sin (\frac{π}{6} x)}{15.8} = \frac{y}{15.8} + \frac{- 40.8}{15.8}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $15.8$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{15.8 \sin (\frac{π}{6} x)}{15.8} = \frac{y}{15.8} + \frac{- 40.8}{15.8}$

Step 3.2.1.2

Divide $\sin (\frac{π}{6} x)$ by $1$ .

$\sin (\frac{π}{6} x) = \frac{y}{15.8} + \frac{- 40.8}{15.8}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply by $1$ .

$\sin (\frac{π}{6} x) = \frac{1 y}{15.8} + \frac{- 40.8}{15.8}$

Step 3.3.1.2

Factor $15.8$ out of $15.8$ .

$\sin (\frac{π}{6} x) = \frac{1 y}{15.8 (1)} + \frac{- 40.8}{15.8}$

Step 3.3.1.3

Separate fractions.

$\sin (\frac{π}{6} x) = \frac{1}{15.8} \cdot \frac{y}{1} + \frac{- 40.8}{15.8}$

Step 3.3.1.4

Divide $1$ by $15.8$ .

$\sin (\frac{π}{6} x) = 0.06329113 \frac{y}{1} + \frac{- 40.8}{15.8}$

Step 3.3.1.5

Divide $y$ by $1$ .

$\sin (\frac{π}{6} x) = 0.06329113 y + \frac{- 40.8}{15.8}$

Step 3.3.1.6

Divide $- 40.8$ by $15.8$ .

$\sin (\frac{π}{6} x) = 0.06329113 y - 2.58227848$

Step 4

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

$\frac{π}{6} x = \arcsin (0.06329113 y - 2.58227848)$

Step 5

Simplify the left side.

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Step 5.1

Combine $\frac{π}{6}$ and $x$ .

$\frac{π x}{6} = \arcsin (0.06329113 y - 2.58227848)$

Step 6

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} \arcsin (0.06329113 y - 2.58227848)$

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Simplify $\frac{6}{π} \cdot \frac{π x}{6}$ .

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Step 7.1.1.1

Cancel the common factor of $6$ .

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Step 7.1.1.1.1

Cancel the common factor.

$\frac{6}{π} \cdot \frac{π x}{6} = \frac{6}{π} \arcsin (0.06329113 y - 2.58227848)$

Step 7.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{6}{π} \arcsin (0.06329113 y - 2.58227848)$

Step 7.1.1.2

Cancel the common factor of $π$ .

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Step 7.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{6}{π} \arcsin (0.06329113 y - 2.58227848)$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{6}{π} \arcsin (0.06329113 y - 2.58227848)$

Step 7.1.1.2.3

Rewrite the expression.

$x = \frac{6}{π} \arcsin (0.06329113 y - 2.58227848)$

Step 7.2

Simplify the right side.

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Step 7.2.1

Combine $\frac{6}{π}$ and $\arcsin (0.06329113 y - 2.58227848)$ .

$x = \frac{6 \arcsin (0.06329113 y - 2.58227848)}{π}$