Trigonometry Examples

$k (x) = \frac{1}{2} \cdot \csc (x)$

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 $x$ .

$0 = \frac{1}{2} \cdot \csc (x)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{1}{2} \cdot \csc (x) = 0$ .

$\frac{1}{2} \cdot \csc (x) = 0$

Step 1.2.2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \csc (x)) = 2 \cdot 0$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $2 (\frac{1}{2} \csc (x))$ .

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Step 1.2.3.1.1.1

Combine $\frac{1}{2}$ and $\csc (x)$ .

$2 \frac{\csc (x)}{2} = 2 \cdot 0$

Step 1.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.2.1

Cancel the common factor.

$2 \frac{\csc (x)}{2} = 2 \cdot 0$

Step 1.2.3.1.1.2.2

Rewrite the expression.

$\csc (x) = 2 \cdot 0$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $2$ by $0$ .

$\csc (x) = 0$

Step 1.2.4

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 $y$ .

$y = \frac{1}{2} \cdot \csc (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{1}{2} \cdot \csc (0)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $\frac{1}{2} \cdot \csc (0)$ .

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Step 2.2.2.1.1

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

$y = \frac{1}{2} \cdot \frac{1}{\sin (0)}$

Step 2.2.2.1.2

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

$y = \frac{1}{2} \cdot \frac{1}{0}$

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