Find the Equation Using Slope-Intercept Form y=(4+csc(x))/(8-csc(x)) , (pi/6,1)

Solution:

Step:1

Determine the $b$ value using the line equation formula.

Step:1.1

Employ the line equation $y = mx + b$ to ascertain $b$.

Step:1.2

Insert the slope $m$ into the equation: $y = \left( \frac{4 + \csc(x)}{8 - \csc(x)} \right) x + b$.

Step:1.3

Plug in the $x$ coordinate: $y = \left( \frac{4 + \csc\left( \frac{\pi}{6} \right)}{8 - \csc\left( \frac{\pi}{6} \right)} \right) \frac{\pi}{6} + b$.

Step:1.4

Input the $y$ coordinate: $1 = \left( \frac{4 + \csc\left( \frac{\pi}{6} \right)}{8 - \csc\left( \frac{\pi}{6} \right)} \right) \frac{\pi}{6} + b$.

Step:1.5

Solve for $b$.

Step:1.5.1

Reformulate the equation: $\left( \frac{4 + \csc\left( \frac{\pi}{6} \right)}{8 - \csc\left( \frac{\pi}{6} \right)} \right) \frac{\pi}{6} + b = 1$.

Step:1.5.2

Simplify the left-hand side.

Step:1.5.2.1

Break down each component.

Step:1.5.2.1.1

Condense the numerator.

Step:1.5.2.1.1.1

$csc\left( \frac{\pi}{6} \right)$ is exactly $2$: $\frac{4 + 2}{8 - \csc\left( \frac{\pi}{6} \right)} \frac{\pi}{6} + b = 1$.

Step:1.5.2.1.1.2

Combine $4$ and $2$: $\frac{6}{8 - \csc\left( \frac{\pi}{6} \right)} \frac{\pi}{6} + b = 1$.

Step:1.5.2.1.2

Condense the denominator.

Step:1.5.2.1.2.1

$csc\left( \frac{\pi}{6} \right)$ is exactly $2$: $\frac{6}{8 - 2} \frac{\pi}{6} + b = 1$.

Step:1.5.2.1.2.2

Multiply $-1$ by $2$: $\frac{6}{8 - 2} \frac{\pi}{6} + b = 1$.

Step:1.5.2.1.2.3

Subtract $2$ from $8$: $\frac{6}{6} \frac{\pi}{6} + b = 1$.

Step:1.5.2.1.3

Eliminate the common $6$ factor.

Step:1.5.2.1.3.1

Remove the common factor: $\frac{\cancel{6}}{\cancel{6}} \frac{\pi}{6} + b = 1$.

Step:1.5.2.1.3.2

Rephrase the equation: $1 \cdot \frac{\pi}{6} + b = 1$.

Step:1.5.2.1.4

Multiply $1$ by $\frac{\pi}{6}$: $\frac{\pi}{6} + b = 1$.

Step:1.5.3

Isolate $b$ by subtracting $\frac{\pi}{6}$ from both sides: $b = 1 - \frac{\pi}{6}$.

Step:2

With the slope $m$ and y-intercept $b$ identified, insert them into $y = mx + b$ to derive the line's equation: $y = \frac{x(4 + \csc(x))}{8 - \csc(x)} + 1 - \frac{\pi}{6}$.

Step:3

The process is complete.

Knowledge Notes:

1. Slope-Intercept Form: The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

2. Cosecant Function: The cosecant ($\csc$) is the reciprocal of the sine function. For an angle $x$, $\csc(x) = \frac{1}{\sin(x)}$. It is undefined when $\sin(x) = 0$.

3. Simplifying Fractions: When simplifying fractions, any common factors in the numerator and denominator can be canceled out.

4. Substitution: To find specific values of a function, substitute the known values of variables into the function.

5. Solving for a Variable: To solve for a variable, isolate it on one side of the equation by performing inverse operations on both sides of the equation.

6. Exact Values of Trigonometric Functions: Some angles have exact trigonometric values that can be used for simplification, such as $\csc\left(\frac{\pi}{6}\right) = 2$.

7. Linear Equations: A linear equation graphs as a straight line on the Cartesian plane and has a constant rate of change or slope.