Trigonometry Examples

$14.55 = 2.35 \sin (0.017 t + 1.86) + 12.2$

Step 1

Rewrite the equation as $2.35 \sin (0.017 t + 1.86) + 12.2 = 14.55$ .

$2.35 \sin (0.017 t + 1.86) + 12.2 = 14.55$

Step 2

Move all terms not containing $\sin (0.017 t + 1.86)$ to the right side of the equation.

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Step 2.1

Subtract $12.2$ from both sides of the equation.

$2.35 \sin (0.017 t + 1.86) = 14.55 - 12.2$

Step 2.2

Subtract $12.2$ from $14.55$ .

$2.35 \sin (0.017 t + 1.86) = 2.35$

Step 3

Divide each term in $2.35 \sin (0.017 t + 1.86) = 2.35$ by $2.35$ and simplify.

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Step 3.1

Divide each term in $2.35 \sin (0.017 t + 1.86) = 2.35$ by $2.35$ .

$\frac{2.35 \sin (0.017 t + 1.86)}{2.35} = \frac{2.35}{2.35}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2.35$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2.35 \sin (0.017 t + 1.86)}{2.35} = \frac{2.35}{2.35}$

Step 3.2.1.2

Divide $\sin (0.017 t + 1.86)$ by $1$ .

$\sin (0.017 t + 1.86) = \frac{2.35}{2.35}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $2.35$ by $2.35$ .

$\sin (0.017 t + 1.86) = 1$

Step 4

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

$0.017 t + 1.86 = \arcsin (1)$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$0.017 t + 1.86 = \frac{π}{2}$

Step 6

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

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Step 6.1

Subtract $1.86$ from both sides of the equation.

$0.017 t = \frac{π}{2} - 1.86$

Step 6.2

Subtract $1.86$ from $\frac{π}{2}$ .

$0.017 t = - 0.28920367$

Step 7

Divide each term in $0.017 t = - 0.28920367$ by $0.017$ and simplify.

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Step 7.1

Divide each term in $0.017 t = - 0.28920367$ by $0.017$ .

$\frac{0.017 t}{0.017} = \frac{- 0.28920367}{0.017}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $0.017$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{0.017 t}{0.017} = \frac{- 0.28920367}{0.017}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 0.28920367}{0.017}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 0.28920367$ by $0.017$ .

$t = - 17.01198077$

Step 8

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.

$0.017 t + 1.86 = π - \frac{π}{2}$

Step 9

Solve for $t$ .

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Step 9.1

Simplify $π - \frac{π}{2}$ .

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Step 9.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$0.017 t + 1.86 = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 9.1.2

Combine fractions.

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Step 9.1.2.1

Combine $π$ and $\frac{2}{2}$ .

$0.017 t + 1.86 = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 9.1.2.2

Combine the numerators over the common denominator.

$0.017 t + 1.86 = \frac{π \cdot 2 - π}{2}$

Step 9.1.3

Simplify the numerator.

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Step 9.1.3.1

Move $2$ to the left of $π$ .

$0.017 t + 1.86 = \frac{2 \cdot π - π}{2}$

Step 9.1.3.2

Subtract $π$ from $2 π$ .

$0.017 t + 1.86 = \frac{π}{2}$

Step 9.2

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

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Step 9.2.1

Subtract $1.86$ from both sides of the equation.

$0.017 t = \frac{π}{2} - 1.86$

Step 9.2.2

Subtract $1.86$ from $\frac{π}{2}$ .

$0.017 t = - 0.28920367$

Step 9.3

Divide each term in $0.017 t = - 0.28920367$ by $0.017$ and simplify.

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Step 9.3.1

Divide each term in $0.017 t = - 0.28920367$ by $0.017$ .

$\frac{0.017 t}{0.017} = \frac{- 0.28920367}{0.017}$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Cancel the common factor of $0.017$ .

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Step 9.3.2.1.1

Cancel the common factor.

$\frac{0.017 t}{0.017} = \frac{- 0.28920367}{0.017}$

Step 9.3.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 0.28920367}{0.017}$

Step 9.3.3

Simplify the right side.

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Step 9.3.3.1

Divide $- 0.28920367$ by $0.017$ .

$t = - 17.01198077$

Step 10

Find the period of $\sin (0.017 t + 1.86)$ .

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

Replace $b$ with $0.017$ in the formula for period.

$\frac{2 π}{| 0.017 |}$

Step 10.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0.017$ is $0.017$ .

$\frac{2 π}{0.017}$

Step 10.4

Replace $π$ with an approximation.

$\frac{2 \cdot 3.14159265}{0.017}$

Step 10.5

Multiply $2$ by $3.14159265$ .

$\frac{6.2831853}{0.017}$

Step 10.6

Divide $6.2831853$ by $0.017$ .

$369.59913571$

Step 11

Add $369.59913571$ to every negative angle to get positive angles.

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Step 11.1

Add $369.59913571$ to $- 17.01198077$ to find the positive angle.

$- 17.01198077 + 369.59913571$

Step 11.2

Subtract $17.01198077$ from $369.59913571$ .

$352.58715493$

Step 11.3

List the new angles.

$t = 352.58715493$

Step 12

The period of the $\sin (0.017 t + 1.86)$ function is $369.59913571$ so values will repeat every $369.59913571$ radians in both directions.

$t = 352.58715493 + 369.59913571 n$ , for any integer $n$