Trigonometry Examples

$20 = 31 \sin (\frac{2 π}{365} x - 1.4)$

Step 1

Rewrite the equation as $31 \sin (\frac{2 π}{365} x - 1.4) = 20$ .

$31 \sin (\frac{2 π}{365} x - 1.4) = 20$

Step 2

Divide each term in $31 \sin (\frac{2 π}{365} x - 1.4) = 20$ by $31$ and simplify.

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Step 2.1

Divide each term in $31 \sin (\frac{2 π}{365} x - 1.4) = 20$ by $31$ .

$\frac{31 \sin (\frac{2 π}{365} x - 1.4)}{31} = \frac{20}{31}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $31$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{31 \sin (\frac{2 π}{365} x - 1.4)}{31} = \frac{20}{31}$

Step 2.2.1.2

Divide $\sin (\frac{2 π}{365} x - 1.4)$ by $1$ .

$\sin (\frac{2 π}{365} x - 1.4) = \frac{20}{31}$

Step 3

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

$\frac{2 π}{365} x - 1.4 = \arcsin (\frac{20}{31})$

Step 4

Simplify the left side.

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Step 4.1

Combine $\frac{2 π}{365}$ and $x$ .

$\frac{2 π x}{365} - 1.4 = \arcsin (\frac{20}{31})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (\frac{20}{31})$ .

$\frac{2 π x}{365} - 1.4 = 0.70123436$

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Add $1.4$ to both sides of the equation.

$\frac{2 π x}{365} = 0.70123436 + 1.4$

Step 6.2

Add $0.70123436$ and $1.4$ .

$\frac{2 π x}{365} = 2.10123436$

Step 7

Multiply both sides of the equation by $\frac{365}{2 π}$ .

$\frac{365}{2 π} \cdot \frac{2 π x}{365} = \frac{365}{2 π} \cdot 2.10123436$

Step 8

Simplify both sides of the equation.

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Step 8.1

Simplify the left side.

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Step 8.1.1

Simplify $\frac{365}{2 π} \cdot \frac{2 π x}{365}$ .

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Step 8.1.1.1

Cancel the common factor of $365$ .

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Step 8.1.1.1.1

Cancel the common factor.

$\frac{365}{2 π} \cdot \frac{2 π x}{365} = \frac{365}{2 π} \cdot 2.10123436$

Step 8.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π x) = \frac{365}{2 π} \cdot 2.10123436$

Step 8.1.1.2

Cancel the common factor of $2 π$ .

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Step 8.1.1.2.1

Factor $2 π$ out of $2 π x$ .

$\frac{1}{2 π} (2 π (x)) = \frac{365}{2 π} \cdot 2.10123436$

Step 8.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π x) = \frac{365}{2 π} \cdot 2.10123436$

Step 8.1.1.2.3

Rewrite the expression.

$x = \frac{365}{2 π} \cdot 2.10123436$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{365}{2 π} \cdot 2.10123436$ .

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Step 8.2.1.1

Multiply $\frac{365}{2 π} \cdot 2.10123436$ .

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Step 8.2.1.1.1

Combine $\frac{365}{2 π}$ and $2.10123436$ .

$x = \frac{365 \cdot 2.10123436}{2 π}$

Step 8.2.1.1.2

Multiply $365$ by $2.10123436$ .

$x = \frac{766.95054307}{2 π}$

Step 8.2.1.2

Replace $π$ with an approximation.

$x = \frac{766.95054307}{2 \cdot 3.14159265}$

Step 8.2.1.3

Multiply $2$ by $3.14159265$ .

$x = \frac{766.95054307}{6.2831853}$

Step 8.2.1.4

Divide $766.95054307$ by $6.2831853$ .

$x = 122.06397003$

Step 9

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.

$\frac{2 π x}{365} - 1.4 = (3.14159265) - 0.70123436$

Step 10

Solve for $x$ .

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Step 10.1

Subtract $0.70123436$ from $3.14159265$ .

$\frac{2 π x}{365} - 1.4 = 2.44035828$

Step 10.2

Move all terms not containing $x$ to the right side of the equation.

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Step 10.2.1

Add $1.4$ to both sides of the equation.

$\frac{2 π x}{365} = 2.44035828 + 1.4$

Step 10.2.2

Add $2.44035828$ and $1.4$ .

$\frac{2 π x}{365} = 3.84035828$

Step 10.3

Multiply both sides of the equation by $\frac{365}{2 π}$ .

$\frac{365}{2 π} \cdot \frac{2 π x}{365} = \frac{365}{2 π} \cdot 3.84035828$

Step 10.4

Simplify both sides of the equation.

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Step 10.4.1

Simplify the left side.

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Step 10.4.1.1

Simplify $\frac{365}{2 π} \cdot \frac{2 π x}{365}$ .

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Step 10.4.1.1.1

Cancel the common factor of $365$ .

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Step 10.4.1.1.1.1

Cancel the common factor.

$\frac{365}{2 π} \cdot \frac{2 π x}{365} = \frac{365}{2 π} \cdot 3.84035828$

Step 10.4.1.1.1.2

Rewrite the expression.

$\frac{1}{2 π} (2 π x) = \frac{365}{2 π} \cdot 3.84035828$

Step 10.4.1.1.2

Cancel the common factor of $2 π$ .

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Step 10.4.1.1.2.1

Factor $2 π$ out of $2 π x$ .

$\frac{1}{2 π} (2 π (x)) = \frac{365}{2 π} \cdot 3.84035828$

Step 10.4.1.1.2.2

Cancel the common factor.

$\frac{1}{2 π} (2 π x) = \frac{365}{2 π} \cdot 3.84035828$

Step 10.4.1.1.2.3

Rewrite the expression.

$x = \frac{365}{2 π} \cdot 3.84035828$

Step 10.4.2

Simplify the right side.

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Step 10.4.2.1

Simplify $\frac{365}{2 π} \cdot 3.84035828$ .

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Step 10.4.2.1.1

Multiply $\frac{365}{2 π} \cdot 3.84035828$ .

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Step 10.4.2.1.1.1

Combine $\frac{365}{2 π}$ and $3.84035828$ .

$x = \frac{365 \cdot 3.84035828}{2 π}$

Step 10.4.2.1.1.2

Multiply $365$ by $3.84035828$ .

$x = \frac{1401.73077548}{2 π}$

Step 10.4.2.1.2

Replace $π$ with an approximation.

$x = \frac{1401.73077548}{2 \cdot 3.14159265}$

Step 10.4.2.1.3

Multiply $2$ by $3.14159265$ .

$x = \frac{1401.73077548}{6.2831853}$

Step 10.4.2.1.4

Divide $1401.73077548$ by $6.2831853$ .

$x = 223.0923818$

Step 11

Find the period of $\sin (\frac{2 π}{365} x - 1.4)$ .

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Step 11.1

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

$\frac{2 π}{| b |}$

Step 11.2

Replace $b$ with $\frac{2 π}{365}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{365} |}$

Step 11.3

$\frac{2 π}{365}$ is approximately $0.0172142$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{365}}$

Step 11.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{365}{2 π}$

Step 11.5

Cancel the common factor of $2 π$ .

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Step 11.5.1

Cancel the common factor.

$2 π \frac{365}{2 π}$

Step 11.5.2

Rewrite the expression.

$365$

Step 12

The period of the $\sin (\frac{2 π}{365} x - 1.4)$ function is $365$ so values will repeat every $365$ radians in both directions.

$x = 122.06397003 + 365 n, 223.0923818 + 365 n$ , for any integer $n$