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Space, shape and measurement: Use the Cartesian co-ordinate system to derive and apply equations

Unit 3: Determine the angle of inclination of a line

Dylan Busa

Unit outcomes

By the end of this unit you will be able to:

  • Use the gradient of a straight line to determine the angle of inclination of the line with the x-axis.
  • Use the angle of inclination to determine the gradient of a straight line.

What you should know

Before you start this unit, make sure you can:

  • Define the tangent ratio.
  • Use a calculator to determine the tangent of an angle.
  • Use a calculator to determine the angle from a given tangent ratio.
  • Use the CAST diagram to identify where tangent is positive and negative.

Refer to level 2 subject outcome 3.6 unit 1 if you need help with any of this.

Introduction

So far in this subject outcome we have been calculating the gradients of straight lines using the idea of a gradient being riserun – the vertical distance between two points on the straight line divided by the horizontal distance between those points. This idea leads us to an equation for gradient that we know well: m=y2y1x2x1.

However, there is another way in which we can measure the gradient of a straight line on the Cartesian plane and this is to measure the angle that the line makes with the x-axis. This is called the angle of inclination.

The angle of inclination

Figure 1 shows what we mean by the angle of inclination. It is the angle that a straight line makes with the positive x-axis.

Figure 1: The angle of inclination

Notice how we always measure the angle of inclination in an anti-clockwise direction from the more positive side of the x-axis of where the line intersects with it.

This means that:

  • the angle of inclination is always 0θ180
  • positive gradients have an angle of inclination 0<θ<90
  • negative gradients have an angle of inclination 90<θ<180.

As the gradient of the straight line changes, so does its angle of inclination. If you have an internet connection visit this practical example.

practical example

Here you will find a slider with which to change the gradient of the line. Change the gradient to see how this affects the angle of inclination of the straight line.

We know that m=y2y1x2x1=ΔyΔx or the change in y divided by the change in x. However, if you look at Figure 2, you will see that, with respect to the angle of inclination, θ, Δy is the opposite side of the triangle and Δx is the adjacent side.

Figure 2:

Therefore, we can say that tanθ=oppositeadjacent=ΔyΔx=y2y1x2x1=m. In other words, tanθ=m for 0θ180.

We know that the tangent function is positive in the first quadrant (0<θ<90) and negative in the second quadrant (90<θ<180). So, it makes sense that lines with an angle of inclination 0<θ<90 have a positive gradient and lines with an angle of inclination 90<θ<180 have a negative gradient.

Angle of inclination and gradient:
m=tanθ

Example 3.1

Determine the gradient of a straight line with an angle of inclination of:

  1. 63
  2. 107

Solution

We know that m=tanθ. Therefore, we need to use a calculator to determine the gradient in each case.
tan63=1.96
tan107=3.27

We can see that the angle of inclination 0<θ<90 results in a positive gradient, while the angle of inclination 90<θ<180 results in a negative gradient.

Example 3.2

Determine the angle of inclination of a straight line with a gradient of:

  1. m=37
  2. m=43

Solution

We know that m=tanθ but in this case, we have been given m and we need to use a calculator to determine the angle of inclination, θ. So, we need to use the tan1 button on our calculator.
tanθ=37θ=23.20
tanθ=43θ=53.13

In this case, the calculator gives an angle in the fourth quadrant. We need to add 180 to put the angle into the second quadrant.
θ=53.13+180=126.87

We can see that positive gradients give angles of inclination 0<θ<90 while negative gradients give angles of inclination 90<θ<180.

Exercise 3.1

  1. Determine the gradient (correct to two decimal places) of straight lines with the following angles of inclination:
    1. 60
    2. 50
    3. 152
    4. 0
    5. 180
    6. 90
  2. Determine the angle of inclination (to two decimal places) of the following straight lines:
    1. 3x+2y6=0
    2. 12yx=3
    3. The line passing through (2,3) and (4,7)
    4. y=6
    5. x=3

The full solutions are at the end of the unit.

Example 3.3

Determine the equation of the straight line passing through (12,14) and perpendicular to a line with an angle of inclination of 71.57.

Solution

The angle of inclination of the other line is 71.57. Therefore, we can find its gradient.
tan71.57=3
m1=3. Therefore, m2=13
yy1=m(xx1)y14=13(x12)y14=13x+16y=13x+2+312y=13x+512

Exercise 3.2

Determine the equation of the straight line perpendicular to the line with an angle of inclination of 165.964 and passing through the point (1,2).

The full solution is at the end of the unit.

Example 3.4

Determine the acute angle (correct to one decimal place) between the line passing through (2,4) and (4,2) and the straight line y=3x+2.

Solution

It will help to draw a sketch of the situation. We can see the angle of inclination of each line (θ and α) and that the acute angle between the two lines is β. If we find θ and α, we can find β using angles in a triangle.

y=3x+2. Therefore, m=3.
tanθ=3θ=71.57

m=y2y1x2x1=4(2)24=66=1
tanα=1α=135

So, the angle next to α inside the triangle is 45 (angles on a straight line are supplementary).

β=1804571.57=63.43 (angles inside a triangle are supplementary).

Exercise 3.3

Determine the angle between the straight lines y=x+3 and yx=12. What can you say about these two straight lines?

The full solution is at the end of the unit.

Summary

In this unit you have learnt the following:

  • The angle of inclination, θ, is the angle made between a straight line and the positive x-axis.
  • The angle of inclination is always 0θ180.
  • Gradient m=tanθ.
  • Positive gradients have an angle of inclination 0θ90.
  • Negative gradients have an angle of inclination 90θ180.

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