Trigonometry Examples

$y = \csc (x) - 8$

Step 1

Set $\csc (x) - 8$ equal to $0$ .

$\csc (x) - 8 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $8$ to both sides of the equation.

$\csc (x) = 8$

Step 2.2

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

$x = arccsc (8)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Evaluate $arccsc (8)$ .

$x = 0.12532783$

Step 2.4

The cosecant 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.

$x = (3.14159265) - 0.12532783$

Step 2.5

Solve for $x$ .

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Step 2.5.1

Remove parentheses.

$x = 3.14159265 - 0.12532783$

Step 2.5.2

Remove parentheses.

$x = (3.14159265) - 0.12532783$

Step 2.5.3

Subtract $0.12532783$ from $3.14159265$ .

$x = 3.01626482$

Step 2.6

Find the period of $\csc (x)$ .

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Step 2.6.1

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

$\frac{2 π}{| b |}$

Step 2.6.2

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

$\frac{2 π}{| 1 |}$

Step 2.6.3

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

$\frac{2 π}{1}$

Step 2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.7

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

$x = 0.12532783 + 2 π n, 3.01626482 + 2 π n$ , for any integer $n$

Step 3