Precalculus Examples

$\sin (- t) = \frac{3}{8}$ , $\csc (t)$

Step 1

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

$- t = \arcsin (\frac{3}{8})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arcsin (\frac{3}{8})$ .

$- t = 0.38439677$

Step 3

Divide each term in $- t = 0.38439677$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- t = 0.38439677$ by $- 1$ .

$\frac{- t}{- 1} = \frac{0.38439677}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} = \frac{0.38439677}{- 1}$

Step 3.2.2

Divide $t$ by $1$ .

$t = \frac{0.38439677}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0.38439677$ by $- 1$ .

$t = - 0.38439677$

Step 4

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.

$- t = (3.14159265) - 0.38439677$

Step 5

Divide each term in $- t = (3.14159265) - 0.38439677$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- t = (3.14159265) - 0.38439677$ by $- 1$ .

$\frac{- t}{- 1} = \frac{3.14159265}{- 1} + \frac{- 0.38439677}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} = \frac{3.14159265}{- 1} + \frac{- 0.38439677}{- 1}$

Step 5.2.2

Divide $t$ by $1$ .

$t = \frac{3.14159265}{- 1} + \frac{- 0.38439677}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Combine the numerators over the common denominator.

$t = \frac{3.14159265 - 0.38439677}{- 1}$

Step 5.3.2

Simplify the expression.

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Step 5.3.2.1

Subtract $0.38439677$ from $3.14159265$ .

$t = \frac{2.75719587}{- 1}$

Step 5.3.2.2

Divide $2.75719587$ by $- 1$ .

$t = - 2.75719587$

Step 6

Find the period of $\sin (- t)$ .

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

Replace $b$ with $- 1$ in the formula for period.

$\frac{2 π}{| - 1 |}$

Step 6.3

The absolute value is the distance between a number and zero. The distance between $- 1$ and $0$ is $1$ .

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

Add $2 π$ to every negative angle to get positive angles.

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Step 7.1

Add $2 π$ to $- 0.38439677$ to find the positive angle.

$- 0.38439677 + 2 π$

Step 7.2

Subtract $0.38439677$ from $2 π$ .

$5.89878853$

Step 7.3

Add $2 π$ to $- 2.75719587$ to find the positive angle.

$- 2.75719587 + 2 π$

Step 7.4

Subtract $2.75719587$ from $2 π$ .

$3.52598942$

Step 7.5

List the new angles.

$t = 3.52598942$

Step 8

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

$t = 3.52598942 + 2 π n, 5.89878853 + 2 π n$ , for any integer $n$

Step 9

Take the base solution.

$t = 3.52598942$

Step 10

Replace the variable $t$ with $3.52598942$ in the expression.

$\csc (3.52598942)$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\csc (3.52598942)$

Decimal Form:

$- 2.\overline{6}$