How do I do this:

Question

645. Find the size of the acute angle formed by the intersecting lines 3x + 2y = 12 and

x − 2y = −2, to the nearest tenth of a degree. Do not use the cosine function and only use the sine or tangent function. Do you need to find the intersection point?

Answer 1

Convert each equation to slope-intercept form

3x+2y = 12

2y = -3x+12

y = (-3/2)x+6

x-2y = -2

-2y = -x-2

y = (1/2)x+1

Given the slopes are not negative reciprocals of each other, then the angle they create is not 90°. To figure out the angle they form, consider dropping the y-intercepts since they have no effect on the angle.

Hence, we have the equations y=(-3/2)x and y=(1/2)x.

To find the acute angle between two lines having slopes m₁, and m₂, respectively, is:

θ = tan^-1[(m₁-m₂)/(1+m₁m₂)]

Use slopes m₁=-3/2 and m₂=1/2:

θ = tan^-1[(m₁-m₂)/(1+m₁m₂)]

θ = tan^-1[(-3/2-1/2)/(1+(-3/2)(1/2))]

θ = tan^-1[-2/(1-3/4)]

θ = tan^-1[-2/(1/4)]

θ = tan^-1(-8)

θ ≈ -82.9°

θ ≈ |-82.9°|

θ ≈ 82.9°

Therefore, the acute angle formed by the intersecting lines is 82.9°

Hope this helped!

Answer 2

Beginning with 3x + 2y = 12, we can put it in slope-intercept form:

2y = -3x + 12

y = (-3/2)x + 6

The same with x − 2y = −2

-2y = -x - 2

y = (1/2)x + 1

We see that these slopes are not negative reciprocals of each other so the angle is not 90°. To determine what the angle is, it may be easier to just drop the y-intercepts of both so that they both go through the origin (the y-intercept does not change the angle between the lines).

So:

y = (-3/2)x

y = (1/2)x

Let α be the angle formed by the x-axis and y = (1/2)x and let β be the angle formed by the x-axis and y = (-3/2)x and θ be the angle formed by y = (1/2)x and y = (-3/2)x

Like this:

Note that we are actually looking for angle z which is the supplement of angle θ

Then tan(α) = 1/2 meaning α = tan-1(1/2) = 26.565°

And tan(β) = -3/2 meaning α = tan-1(-3/2) = -56.310°

So θ = α + |β| = 26.565 + 56.310 = 82.875°

That makes angle z = 180 - 82.875° = 97.1°

Answer 3

Draw the two lines.

Let the x-axis be a third side of the triangle with vertices (-2, 0), (4, 0), and (7, - 4.5)

The slope of the line at (-2, 0) is 1/2. Therefore the sine of the angle at that vertex is 1/2.

The angle at vertex (-2, 0) is 30°.

Now use Law of Sines to determine size of angle at vertex (7, -4.5)