Precalculus Examples

$y = 6 \csc (x)$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 6 \csc (x)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $6 \csc (x) = 0$ .

$6 \csc (x) = 0$

Step 1.2.2

Divide each term in $6 \csc (x) = 0$ by $6$ and simplify.

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Step 1.2.2.1

Divide each term in $6 \csc (x) = 0$ by $6$ .

$\frac{6 \csc (x)}{6} = \frac{0}{6}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{6 \csc (x)}{6} = \frac{0}{6}$

Step 1.2.2.2.1.2

Divide $\csc (x)$ by $1$ .

$\csc (x) = \frac{0}{6}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $6$ .

$\csc (x) = 0$

Step 1.2.3

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 6 \csc (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 6 \csc (0)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $6 \csc (0)$ .

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Step 2.2.2.1.1

Rewrite $\csc (0)$ in terms of sines and cosines.

$y = 6 \frac{1}{\sin (0)}$

Step 2.2.2.1.2

The exact value of $\sin (0)$ is $0$ .

$y = 6 (\frac{1}{0})$

Step 2.2.2.2

The equation cannot be solved because it is undefined.

$Undefined$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 4