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Special Slopes and Sign
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.
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Special Slopes and Sign
What is it?
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.
In simple words
- 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
Important terms
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).
What you should learn
- 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
Recommended for Class 9
Full school-level explanation
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.
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.
Remember
- 45 deg
- m=1
- 60
- sqrt(3)
- 30
- 1/sqrt(3)
- positive slope
- 0 < theta < 90
Deeper understanding
Go beyond memorising Special Slopes and Sign. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
Core Idea
Key Terms
Main Pattern
Worked Reminder
Exam focus
Quick revision
- 45 deg
- m=1
- 60
- sqrt(3)
- 45 deg: At 45 deg, tan 45 = 1 so m=1.
- m=1: At 45 deg, tan 45 = 1 so m=1.
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.
For long answers on Special Slopes and Sign: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- 45 deg
- m=1
- 60
- sqrt(3)
Important formulas
- 45 deg
- m=1
- 60
Common exam traps
- 45 deg gives m=1; 60 deg gives sqrt(3); 30 deg gives 1/sqrt(3).
- Positive slope means 0 < theta < 90.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
45 deg gives m=1; 60 deg gives sqrt(3); 30 deg gives 1/sqrt(3).
- Mistake
Positive slope means 0 < theta < 90.
- Mistake
Do not copy a formula blindly; check that its conditions and units match the question.
- 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
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
45 deg
m=1
60
sqrt(3)
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
m=1
- Definition
- 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
- Formula
- m=1
- Critical warning
- 45 deg gives m=1; 60 deg gives sqrt(3); 30 deg gives 1/sqrt(3).
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Exam Practice
Define
Q1. Define Special Slopes and Sign.
Explain
Q2. Explain: What it means.
Example
Q3. Give one clear example of Special Slopes and Sign and say why it fits.
Mistake
Q4. State one common mistake students make about Special Slopes and Sign.
Exam note
Q5. Write a short exam note on Special Slopes and Sign: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.