Chapter 6, Problem 1RE

To determine

To calculate: The value of the inclination $\theta$ (in radians and degree) for the line which passes through the points $\left(-1,2\right)$ and $\left(2,5\right)$.

Answer to Problem 1RE

The value of the inclination $\theta$ (in radians) is $\frac{\pi }{4}$ and the value of the inclination $\theta$ (in degrees) is $45°$.

Explanation of Solution

Calculation:

Consider the provided points,

$\left(-1,2\right)$ and $\left(2,5\right)$

Let $\left(-1,2\right)=\left(x_1,y_1\right)$ and $\left(2,5\right)=\left(x_2,y_2\right)$

The value of the inclination passing through two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ can be found by using the formula for the slope,

$m=\frac{y_2-y_1}{x_2-x_1}$

Substitute $5$ for $y_2$, $2$ for $y_1$, $2$ for $x_2$, and $-1$ for $x_1$,

$\begin{array}{c}m=\frac{5-2}{2-\left(-1\right)}\\ =\frac{3}{2+1}\\ =\frac{3}{3}\\ =1\end{array}$

The inclination of a line is represented by,

$\tan \theta =m$

Substitute $1$ for $m$,

$\begin{array}{c}\tan \theta =1\\ \theta =\arctan \left(1\right)\\ =\frac{\pi }{4}\end{array}$

Convert radians into degrees,

$\begin{array}{c}\theta =\frac{\pi }{4}\text{ rad}\\ =\left(\frac{\cancel{\pi }}{4}\text{ra<menclose notation="updiagonalstrike">d</menclose>}\right)\left(\frac{180\text{ deg}}{\cancel{\pi }\text{ ra<menclose notation="updiagonalstrike">d</menclose>}}\right)\\ =45°\end{array}$

The graph of the line passing through the points $\left(-1,2\right)$ and $\left(2,5\right)$ is shown below,

Hence, the value of the inclination $\theta$ (in radians) is $\frac{\pi }{4}$ and the value of the inclination $\theta$ (in degrees) is $45°$.