To find the equation of the straight line that meets the given criteria, let's break down the problem step-by-step.
1. Identify the Given Information:
- The y-intercept of the line: -2
- The angle the line makes with the x-axis: [tex]\(\tan^{-1} \left( \frac{3}{4} \right)\)[/tex]
2. Determine the Slope of the Line:
- The slope [tex]\(m\)[/tex] of a line inclined at an angle [tex]\(\theta\)[/tex] with the x-axis is given by the tangent of the angle, i.e., [tex]\(m = \tan(\theta)\)[/tex].
- Here, [tex]\(\theta = \tan^{-1} \left(\frac{3}{4}\right)\)[/tex].
- Since [tex]\(\tan(\tan^{-1} (x)) = x\)[/tex], we have [tex]\(m = \frac{3}{4}\)[/tex].
3. Write the General Equation of the Line:
- The equation of a straight line in slope-intercept form is [tex]\(y = mx + c\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept.
- Given: [tex]\(m = \frac{3}{4}\)[/tex] and [tex]\(c = -2\)[/tex].
4. Substitute the Known Values:
- Substitute [tex]\(m = \frac{3}{4}\)[/tex] and [tex]\(c = -2\)[/tex] into the slope-intercept form.
- [tex]\(y = \frac{3}{4}x - 2\)[/tex].
5. Final Equation:
- The equation of the line is [tex]\(y = 0.75x + (-2)\)[/tex].
- Simplifying further:
[tex]\[
y = 0.75x - 2
\][/tex]
So, the equation of the straight line that cuts off an intercept of -2 from the y-axis and is inclined at an angle of [tex]\(\tan^{-1} \left(\frac{3}{4}\right)\)[/tex] with the x-axis is:
[tex]\[
y = 0.75x - 2
\][/tex]
In summary:
- The slope of the line is 0.75.
- The y-intercept is -2.
- The equation of the line is [tex]\(y = 0.75x - 2\)[/tex].
