MathematicsChapter 8: Geometry of Straight Lines
Special Slopes and Sign
A few inclinations have exact gradients you should know.
Learning objectives
- Explain special slopes and sign in clear mathematical language
- Use the main rules and formulas in worked examples
- Recognise and correct common errors in this topic
What this grid covers
A few inclinations have exact gradients you should know. At 45 deg, tan 45 = 1 so m=1. At 60 deg, tan 60 = sqrt(3). At 30 deg, tan 30 = 1/sqrt(3). Sign follows the angle: positive slope when 0 < theta < 90 (the line rises left to right). Negative slope when 90 < theta < 180 (the line falls left to right). Horizontal m=0 sits at the boundary theta=0; vertical is excluded.
Where / symbols
- 45 deg
- At 45 deg, tan 45 = 1 so m=1.
- m=1
- At 45 deg, tan 45 = 1 so m=1.
- 60
- At 60 deg, tan 60 = sqrt(3).
- sqrt(3)
- At 60 deg, tan 60 = sqrt(3).
Relationship
- m=1
What it means
A few inclinations have exact gradients you should know.
Main ideas
0 deg (horizontal): m = 0; Flat; no rise 30 deg: m = 1/sqrt(3); Gentle rise, positive 45 deg: m = 1; Equal rise and run 60 deg: m = sqrt(3); Steep rise, positive
Why it matters
This method connects the definition to worked questions and later mathematics.
How to answer in an exam
45 deg gives m=1; 60 deg gives sqrt(3); 30 deg gives 1/sqrt(3). Positive slope means 0 < theta < 90. Negative slope means 90 < theta < 180. Do not say a falling line has theta negative; inclination is still anti-clockwise from the positive x-axis, so it is obtuse.
Link theta, tan theta, and the look of the line:
| Inclination theta | Gradient m = tan theta | What the line does |
|---|---|---|
| 1. 0 deg (horizontal) | m = 0 | Flat; no rise |
| 2. 30 deg | m = 1/sqrt(3) | Gentle rise, positive |
| 3. 45 deg | m = 1 | Equal rise and run |
| 4. 60 deg | m = sqrt(3) | Steep rise, positive |
| 5. 90 deg (vertical) | m undefined | No run; not a function y=mx+c |
| 6. Between 90 deg and 180 deg | m negative (tan theta < 0) | Falls left to right; 90 < theta < 180 |
In simple words
45 deg gives m=1; 60 deg gives sqrt(3); 30 deg gives 1/sqrt(3). Positive slope means 0 < theta < 90. Negative slope means 90 < theta < 180. Do not say a falling line has theta negative; inclination is still anti-clockwise from the positive x-axis, so it is obtuse.
Key points
- 45 deg
- m=1
- 60
- sqrt(3)
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Special Slopes and Sign
Sticky Notes · Quick Revision
Core Idea
- A few inclinations have exact gradients you should know.
Key Terms
- 45 deg: At 45 deg, tan 45 = 1 so m=1.
- m=1: At 45 deg, tan 45 = 1 so m=1.
Main Pattern
- 0 deg (horizontal): m = 0
- 30 deg: m = 1/sqrt(3)
Worked Reminder
- 0 deg (horizontal): m = 0; Flat; no rise
- 30 deg: m = 1/sqrt(3); Gentle rise, positive
- 45 deg: m = 1; Equal rise and run
Common Mistake
- 45 deg gives m=1; 60 deg gives sqrt(3); 30 deg gives 1/sqrt(3). Positive slope means 0 < theta < 90. Negative slope means 90 < theta < 180. Do not say a falling line has theta negative; inclination is still anti-clockwise from the positive x-axis, so it is obtuse.
Exam Tip
- State the rule for special slopes and sign, show each algebraic step, and check the final answer against the question.
60-second revision
- A few inclinations have exact gradients you should know.
- 0 deg (horizontal): m = 0; Flat; no rise
- 30 deg: m = 1/sqrt(3); Gentle rise, positive
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