b The acute angle formed by the line and the x-axis is also an acute angle of the slope triangle.
A
aExample Solution: tan^(- 1) 31; about 71.6^(∘)
B
bExample Solution: tan^(- 1) 43; about 53.1^(∘)
Practice makes perfect
a We are asked to write an expression that can be used to find the measure of the acute angle formed by the line y=3x and the x-axis. Let's star by sketching the line. To do so we will write it in slope-intercept form.
y=3x ⇔ y= 3x+
The slope of the line is 3 and its y-intercept is . Let's sketch the line!
We can see that the acute angle formed by the line and the x-axis is also an acute angle of the slope triangle. Let a represent the angle measure of this acute angle.
Since the slope triangle is a right triangle and we know the lengths of its legs, we can write the tangent of the angle a. The tangent of a is the ratio of the opposite leg to the adjacent leg.
tana=Opposite leg/Adjacent leg ⇔ tana=3/1
Note that the tangent of the angle formed by the line and the x-axis on the right-hand side of the x-intercept is equal to the slope of the line. Remember that this is true for any line. To find the angle measure of the angle a formed by the line and the x-axis, we will use the inverse tangent ratio.
tana=3/1 ⇔ a=tan^(- 1)3/1
Let's approximate the inverse tangent of 31 to the nearest tenth.
Using the expression tan^(- 1) 31, we have found that the acute angle formed by the line y=3x and the x-axis is about 71.6^(∘).
b We are asked to write an expression that can be used to find the measure of the acute angle formed by the line y= 43x+4 and the x-axis. Let's start by sketching the line. To do it, we will highlight the slope m and the y-intercept b of the line.
y= 4/3x+4
The slope of the line is 43 and its y-intercept is 4. Let's sketch the line!
Recall that in Part A we noticed that the acute angle formed by the line and the x-axis is an acute angle of the slope triangle at the same time. This is also true for the line y= 43x+4. Again, let a represent the angle measure of this acute angle.
In Part A we also said that the tangent of any angle formed by a line and the x-axis on the right-hand side of the x-intercept is equal to the slope of the line. Therefore, the tangent of a is equal to 43.
tana=4/3
To find the angle measure of the angle a formed by the line and the x-axis, we will use the inverse tangent ratio.
tana=4/3 ⇔ a=tan^(- 1)4/3
Finally, let's approximate the inverse tangent of 43 to the nearest tenth.
