Calculus Examples

$y = - 8 \csc (3 x)$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

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

$0 = - 8 \csc (3 x)$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $- 8 \csc (3 x) = 0$ .

$- 8 \csc (3 x) = 0$

Step 1.2.2

Divide each term in $- 8 \csc (3 x) = 0$ by $- 8$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $- 8 \csc (3 x) = 0$ by $- 8$ .

$\frac{- 8 \csc (3 x)}{- 8} = \frac{0}{- 8}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 8 \csc (3 x)}{- 8} = \frac{0}{- 8}$

Step 1.2.2.2.1.2

Divide $\csc (3 x)$ by $1$ .

$\csc (3 x) = \frac{0}{- 8}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Divide $0$ by $- 8$ .

$\csc (3 x) = 0$

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.

Tap for more steps...

Step 2.1

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

$y = - 8 \csc (3 (0))$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = - 8 \csc (3 (0))$

Step 2.2.2

Simplify the right side.

Tap for more steps...

Step 2.2.2.1

Simplify $- 8 \csc (3 (0))$ .

Tap for more steps...

Step 2.2.2.1.1

Rewrite $\csc (3 (0))$ in terms of sines and cosines.

$y = - 8 \frac{1}{\sin (3 (0))}$

Step 2.2.2.1.2

Multiply $3$ by $0$ .

$y = - 8 \frac{1}{\sin (0)}$

Step 2.2.2.1.3

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

$y = - 8 (\frac{1}{0})$

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