Precalc Problem - Angle of Inclination?

Question

Please help me with this problem:

Find the angle of inclination for the line 2x+y=9 to the nearest degree.

Answer 1

The angle of inclination is how much a line is tilted up or down when looking at it from left to right. In a real world scenario, if you were looking up at the top of a building, your angle of inclination (aka angle of elevation) would be formed by the angle between the ground and your line of sight to the top of the building. In looking at lines, you'd be measuring from the x-axis to the line.

Your line, when graphed, actually points downward, so you're going to have a negative angle of inclination (aka angle of declination). You can find that angle by first finding the slope of the line, and then using some reverse trig to find the measurement of the angle.

Let's start with the slope issue. Rewrite the equation in slope-intercept form to read off the slope:

2x + y = 9
y = -2x + 9

The slope is then -2. Recall that slope is rise over run, or dy/dx. If you visualize that as the legs in a triangle, you'd have one side of your triangle being 1 (short horizontal side), another side being 2 (long vertical side), and your hypotenuse being part of the line itself (we're dropping the negative sign for now). The angle we're looking for is the one adjacent to the short horizontal side. Since we have the sides opposite and adjacent to the angle, we can use an inverse tangent function to figure out the angle:

tan (theta) = y/x
tan (theta) = 2/1
theta ~ 63.43 degrees

Now since your angle is actually sloping down, your angle of inclination will be -63.43 degrees, or rounded to the nearest degree, -63 degrees.

Answer 2

Step 1: Convert to slope-intercept and find slope.
2x+y=9
7 = 9 - 2x
slope = -2.

Convert slope to trigonometric measure:
Rise = opposite, run = adjacent, so slope = tan x
tan x = -2
Plug into calculator:
x = -63 degrees