Trigonometry Examples

$4 = - 5 \sin (\frac{3 π}{4} t)$

Step 1

Rewrite the equation as $- 5 \sin (\frac{3 π}{4} t) = 4$ .

$- 5 \sin (\frac{3 π}{4} t) = 4$

Step 2

Divide each term in $- 5 \sin (\frac{3 π}{4} t) = 4$ by $- 5$ and simplify.

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Step 2.1

Divide each term in $- 5 \sin (\frac{3 π}{4} t) = 4$ by $- 5$ .

$\frac{- 5 \sin (\frac{3 π}{4} t)}{- 5} = \frac{4}{- 5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 5$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 5 \sin (\frac{3 π}{4} t)}{- 5} = \frac{4}{- 5}$

Step 2.2.1.2

Divide $\sin (\frac{3 π}{4} t)$ by $1$ .

$\sin (\frac{3 π}{4} t) = \frac{4}{- 5}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$\sin (\frac{3 π}{4} t) = - \frac{4}{5}$

Step 3

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

$\frac{3 π}{4} t = \arcsin (- \frac{4}{5})$

Step 4

Simplify the left side.

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Step 4.1

Combine $\frac{3 π}{4}$ and $t$ .

$\frac{3 π t}{4} = \arcsin (- \frac{4}{5})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arcsin (- \frac{4}{5})$ .

$\frac{3 π t}{4} = - 0.92729521$

Step 6

Multiply both sides of the equation by $\frac{4}{3 π}$ .

$\frac{4}{3 π} \cdot \frac{3 π t}{4} = \frac{4}{3 π} \cdot - 0.92729521$

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 $\frac{4}{3 π} \cdot \frac{3 π t}{4}$ .

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Step 7.1.1.1

Cancel the common factor of $4$ .

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Step 7.1.1.1.1

Cancel the common factor.

$\frac{4}{3 π} \cdot \frac{3 π t}{4} = \frac{4}{3 π} \cdot - 0.92729521$

Step 7.1.1.1.2

Rewrite the expression.

$\frac{1}{3 π} (3 π t) = \frac{4}{3 π} \cdot - 0.92729521$

Step 7.1.1.2

Cancel the common factor of $3 π$ .

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Step 7.1.1.2.1

Factor $3 π$ out of $3 π t$ .

$\frac{1}{3 π} (3 π (t)) = \frac{4}{3 π} \cdot - 0.92729521$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{1}{3 π} (3 π t) = \frac{4}{3 π} \cdot - 0.92729521$

Step 7.1.1.2.3

Rewrite the expression.

$t = \frac{4}{3 π} \cdot - 0.92729521$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{4}{3 π} \cdot - 0.92729521$ .

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Step 7.2.1.1

Multiply $\frac{4}{3 π} \cdot - 0.92729521$ .

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Step 7.2.1.1.1

Combine $\frac{4}{3 π}$ and $- 0.92729521$ .

$t = \frac{4 \cdot - 0.92729521}{3 π}$

Step 7.2.1.1.2

Multiply $4$ by $- 0.92729521$ .

$t = \frac{- 3.70918087}{3 π}$

Step 7.2.1.2

Move the negative in front of the fraction.

$t = - \frac{3.70918087}{3 π}$

Step 7.2.1.3

Replace $π$ with an approximation.

$t = - \frac{3.70918087}{3 \cdot 3.14159265}$

Step 7.2.1.4

Multiply $3$ by $3.14159265$ .

$t = - \frac{3.70918087}{9.42477796}$

Step 7.2.1.5

Divide $3.70918087$ by $9.42477796$ .

$t = - 1 \cdot 0.39355631$

Step 7.2.1.6

Multiply $- 1$ by $0.39355631$ .

$t = - 0.39355631$

Step 8

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$\frac{3 π t}{4} = 2 (3.14159265) + 0.92729521 + 3.14159265$

Step 9

Simplify the expression to find the second solution.

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Step 9.1

Subtract $2 π$ from $2 (3.14159265) + 0.92729521 + 3.14159265$ .

$\frac{3 π t}{4} = 2 (3.14159265) + 0.92729521 + 3.14159265 - 2 π$

Step 9.2

The resulting angle of $4.06888787$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.92729521 + 3.14159265$ .

$\frac{3 π t}{4} = 4.06888787$

Step 9.3

Solve for $t$ .

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Step 9.3.1

Multiply both sides of the equation by $\frac{4}{3 π}$ .

$\frac{4}{3 π} \cdot \frac{3 π t}{4} = \frac{4}{3 π} \cdot 4.06888787$

Step 9.3.2

Simplify both sides of the equation.

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Step 9.3.2.1

Simplify the left side.

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Step 9.3.2.1.1

Simplify $\frac{4}{3 π} \cdot \frac{3 π t}{4}$ .

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Step 9.3.2.1.1.1

Cancel the common factor of $4$ .

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Step 9.3.2.1.1.1.1

Cancel the common factor.

$\frac{4}{3 π} \cdot \frac{3 π t}{4} = \frac{4}{3 π} \cdot 4.06888787$

Step 9.3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{3 π} (3 π t) = \frac{4}{3 π} \cdot 4.06888787$

Step 9.3.2.1.1.2

Cancel the common factor of $3 π$ .

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Step 9.3.2.1.1.2.1

Factor $3 π$ out of $3 π t$ .

$\frac{1}{3 π} (3 π (t)) = \frac{4}{3 π} \cdot 4.06888787$

Step 9.3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{3 π} (3 π t) = \frac{4}{3 π} \cdot 4.06888787$

Step 9.3.2.1.1.2.3

Rewrite the expression.

$t = \frac{4}{3 π} \cdot 4.06888787$

Step 9.3.2.2

Simplify the right side.

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Step 9.3.2.2.1

Simplify $\frac{4}{3 π} \cdot 4.06888787$ .

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Step 9.3.2.2.1.1

Multiply $\frac{4}{3 π} \cdot 4.06888787$ .

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Step 9.3.2.2.1.1.1

Combine $\frac{4}{3 π}$ and $4.06888787$ .

$t = \frac{4 \cdot 4.06888787}{3 π}$

Step 9.3.2.2.1.1.2

Multiply $4$ by $4.06888787$ .

$t = \frac{16.27555148}{3 π}$

Step 9.3.2.2.1.2

Replace $π$ with an approximation.

$t = \frac{16.27555148}{3 \cdot 3.14159265}$

Step 9.3.2.2.1.3

Multiply $3$ by $3.14159265$ .

$t = \frac{16.27555148}{9.42477796}$

Step 9.3.2.2.1.4

Divide $16.27555148$ by $9.42477796$ .

$t = 1.72688964$

Step 10

Find the period of $\sin (\frac{3 π}{4} t)$ .

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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 $\frac{3 π}{4}$ in the formula for period.

$\frac{2 π}{| \frac{3 π}{4} |}$

Step 10.3

$\frac{3 π}{4}$ is approximately $2.35619449$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3 π}{4}}$

Step 10.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{4}{3 π}$

Step 10.5

Cancel the common factor of $π$ .

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Step 10.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{4}{3 π}$

Step 10.5.2

Factor $π$ out of $3 π$ .

$π \cdot 2 \frac{4}{π \cdot 3}$

Step 10.5.3

Cancel the common factor.

$π \cdot 2 \frac{4}{π \cdot 3}$

Step 10.5.4

Rewrite the expression.

$2 (\frac{4}{3})$

Step 10.6

Combine $2$ and $\frac{4}{3}$ .

$\frac{2 \cdot 4}{3}$

Step 10.7

Multiply $2$ by $4$ .

$\frac{8}{3}$

Step 11

Add $\frac{8}{3}$ to every negative angle to get positive angles.

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Step 11.1

Add $\frac{8}{3}$ to $- 0.39355631$ to find the positive angle.

$- 0.39355631 + \frac{8}{3}$

Step 11.2

To write $- 0.39355631$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} - 0.39355631 \cdot \frac{3}{3}$

Step 11.3

Combine fractions.

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Step 11.3.1

Combine $- 0.39355631$ and $\frac{3}{3}$ .

$\frac{8}{3} + \frac{- 0.39355631 \cdot 3}{3}$

Step 11.3.2

Combine the numerators over the common denominator.

$\frac{8 - 0.39355631 \cdot 3}{3}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Multiply $- 0.39355631$ by $3$ .

$\frac{8 - 1.18066894}{3}$

Step 11.4.2

Subtract $1.18066894$ from $8$ .

$\frac{6.81933105}{3}$

Step 11.5

Divide $6.81933105$ by $3$ .

$2.27311035$

Step 11.6

List the new angles.

$t = 2.27311035$

Step 12

The period of the $\sin (\frac{3 π}{4} t)$ function is $\frac{8}{3}$ so values will repeat every $\frac{8}{3}$ radians in both directions.

$t = 1.72688964 + \frac{8 n}{3}, 2.27311035 + \frac{8 n}{3}$ , for any integer $n$