Trigonometry Examples

$2.25 = - 0.75 \sin (2 π t) + 2.5$

Step 1

Rewrite the equation as $- 0.75 \sin (2 π t) + 2.5 = 2.25$ .

$- 0.75 \sin (2 π t) + 2.5 = 2.25$

Step 2

Move all terms not containing $\sin (2 π t)$ to the right side of the equation.

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Step 2.1

Subtract $2.5$ from both sides of the equation.

$- 0.75 \sin (2 π t) = 2.25 - 2.5$

Step 2.2

Subtract $2.5$ from $2.25$ .

$- 0.75 \sin (2 π t) = - 0.25$

Step 3

Divide each term in $- 0.75 \sin (2 π t) = - 0.25$ by $- 0.75$ and simplify.

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Step 3.1

Divide each term in $- 0.75 \sin (2 π t) = - 0.25$ by $- 0.75$ .

$\frac{- 0.75 \sin (2 π t)}{- 0.75} = \frac{- 0.25}{- 0.75}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 0.75$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 0.75 \sin (2 π t)}{- 0.75} = \frac{- 0.25}{- 0.75}$

Step 3.2.1.2

Divide $\sin (2 π t)$ by $1$ .

$\sin (2 π t) = \frac{- 0.25}{- 0.75}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 0.25$ by $- 0.75$ .

$\sin (2 π t) = 0.\overline{3}$

Step 4

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

$2 π t = \arcsin (0.\overline{3})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (0.\overline{3})$ .

$2 π t = 0.3398369$

Step 6

Divide each term in $2 π t = 0.3398369$ by $2 π$ and simplify.

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Step 6.1

Divide each term in $2 π t = 0.3398369$ by $2 π$ .

$\frac{2 π t}{2 π} = \frac{0.3398369}{2 π}$

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.

$\frac{2 π t}{2 π} = \frac{0.3398369}{2 π}$

Step 6.2.1.2

Rewrite the expression.

$\frac{π t}{π} = \frac{0.3398369}{2 π}$

Step 6.2.2

Cancel the common factor of $π$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{π t}{π} = \frac{0.3398369}{2 π}$

Step 6.2.2.2

Divide $t$ by $1$ .

$t = \frac{0.3398369}{2 π}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Replace $π$ with an approximation.

$t = \frac{0.3398369}{2 \cdot 3.14159265}$

Step 6.3.2

Multiply $2$ by $3.14159265$ .

$t = \frac{0.3398369}{6.2831853}$

Step 6.3.3

Divide $0.3398369$ by $6.2831853$ .

$t = 0.05408672$

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.

$2 π t = (3.14159265) - 0.3398369$

Step 8

Solve for $t$ .

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Step 8.1

Subtract $0.3398369$ from $3.14159265$ .

$2 π t = 2.80175574$

Step 8.2

Divide each term in $2 π t = 2.80175574$ by $2 π$ and simplify.

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Step 8.2.1

Divide each term in $2 π t = 2.80175574$ by $2 π$ .

$\frac{2 π t}{2 π} = \frac{2.80175574}{2 π}$

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.

$\frac{2 π t}{2 π} = \frac{2.80175574}{2 π}$

Step 8.2.2.1.2

Rewrite the expression.

$\frac{π t}{π} = \frac{2.80175574}{2 π}$

Step 8.2.2.2

Cancel the common factor of $π$ .

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Step 8.2.2.2.1

Cancel the common factor.

$\frac{π t}{π} = \frac{2.80175574}{2 π}$

Step 8.2.2.2.2

Divide $t$ by $1$ .

$t = \frac{2.80175574}{2 π}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Replace $π$ with an approximation.

$t = \frac{2.80175574}{2 \cdot 3.14159265}$

Step 8.2.3.2

Multiply $2$ by $3.14159265$ .

$t = \frac{2.80175574}{6.2831853}$

Step 8.2.3.3

Divide $2.80175574$ by $6.2831853$ .

$t = 0.44591327$

Step 9

Find the period of $\sin (2 π t)$ .

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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 $2 π$ in the formula for period.

$\frac{2 π}{| 2 π |}$

Step 9.3

$2 π$ is approximately $6.2831853$ which is positive so remove the absolute value

$\frac{2 π}{2 π}$

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Cancel the common factor.

$\frac{2 π}{2 π}$

Step 9.4.2

Rewrite the expression.

$\frac{π}{π}$

Step 9.5

Cancel the common factor of $π$ .

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Step 9.5.1

Cancel the common factor.

$\frac{π}{π}$

Step 9.5.2

Rewrite the expression.

$1$

Step 10

The period of the $\sin (2 π t)$ function is $1$ so values will repeat every $1$ radians in both directions.

$t = 0.05408672 + n, 0.44591327 + n$ , for any integer $n$