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More Allied Forms (270, Negative)
sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A, and the rest as the book table. Meaning: even/odd plus CAST covers almost every allied question. Learn a table or derive from 180 and 90.
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More Allied Forms (270, Negative)
What is it?
sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A, and the rest as the book table. Meaning: even/odd plus CAST covers almost every allied question. Learn a table or derive from 180 and 90.
In simple words
- sin(360-A)=-sin A, cos(360-A)=cos A.
- sin(-A)=-sin A (odd), cos(-A)=cos A (even).
- sin(270-A)=-cos A, and the rest as the book table.
- Cosine is even: negatives do not flip cos. Sine does flip.
Important terms
MORE ALLIED FORMS (270, NEGATIVE): sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A, and the rest as the book table. Meaning: even/odd plus CAST covers almost every allied question. Learn a table or derive from 180 and 90.
1. NEGATIVE ANGLES: sin(-30)=-12. cos(-30)=sqrt(3)/2. tan(-A)=-tan A. Useful when Arg was negative in complex polar form too.
2. 270 AND 360: 270=180+90, so use two steps if you forget the table. 360-A: like -A for sine and cosine of a positive A between 0 and 360.
What you should learn
- State the main ideas and notation used in more allied forms (270, negative).
- Apply the rules of more allied forms (270, negative) in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- sin(-A)=-sin A
- cos(-A)=cos A
- tan odd
NEGATIVE ANGLES
sin(-30)=-12. cos(-30)=sqrt(3)/2. tan(-A)=-tan A. Useful when Arg was negative in complex polar form too.
270 AND 360
270=180+90, so use two steps if you forget the table. 360-A: like -A for sine and cosine of a positive A between 0 and 360.
Recommended for Class 11
Full school-level explanation
sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A, and the rest as the book table. Meaning: even/odd plus CAST covers almost every allied question. Learn a table or derive from 180 and 90.
Even/odd, then 270:
WHAT — MORE ALLIED FORMS (270, NEGATIVE)
sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A, and the rest as the book table. Meaning: even/odd plus CAST covers almost every allied question. Learn a table or derive from 180 and 90.
RULE — 1. NEGATIVE ANGLES
sin(-30)=-12. cos(-30)=sqrt(3)/2. tan(-A)=-tan A. Useful when Arg was negative in complex polar form too.
METHOD — 2. 270 AND 360
270=180+90, so use two steps if you forget the table. 360-A: like -A for sine and cosine of a positive A between 0 and 360.
WORKED EXAMPLE
Worked: cos(-60). cos even: cos(-60)=cos 60 cos 60 = 12 not -12 (that would be sine)
NEGATIVE ANGLES
sin(-30)=-12. cos(-30)=sqrt(3)/2. tan(-A)=-tan A. Useful when Arg was negative in complex polar form too.
Remember
- odd / even
Examples
- sin(-A)=-sin A
- cos(-A)=cos A
- tan odd
270 AND 360
270=180+90, so use two steps if you forget the table. 360-A: like -A for sine and cosine of a positive A between 0 and 360.
Remember
- two-step if needed
Examples
- sin(270-A)=-cos A school table
- cos(360-A)=cos A
- Do not mix with 180-A
How to answer a more allied forms (270, negative) question
How it works
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Remember
- sin odd, cos even
- 360-A is a fourth-quadrant picture
- 12
- cos even
- odd / even
- two-step if needed
- sin odd, cos even
- 360-A is a fourth-quadrant picture
How the ideas connect
Worked: cos(-60). cos even: cos(-60)=cos 60 cos 60 = 12 not -12 (that would be sine)
Deeper understanding
Go beyond memorising More Allied Forms (270, Negative). Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: cos(-60). cos even: cos(-60)=cos 60 cos 60 = 12 not -12 (that would be sine)
Common mistakes
Exam focus
Core Idea
1. NEGATIVE ANGLES
2. 270 AND 360
Worked Example
Exam focus
Quick revision
- sin odd, cos even
- 360-A is a fourth-quadrant picture
- 12
- cos even
- MORE ALLIED FORMS (270, NEGATIVE): sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A,
- 1. NEGATIVE ANGLES: sin(-30)=-12. cos(-30)=sqrt(3)/2. tan(-A)=-tan A. Useful when Arg was negative in complex polar for
Cosine is even: negatives do not flip cos. Sine does flip.
For long answers on More Allied Forms (270, Negative): define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- sin odd, cos even
- 360-A is a fourth-quadrant picture
- 12
- cos even
Important formulas
- sin odd, cos even
- 360-A is a fourth-quadrant picture
- 12
Common exam traps
- Cosine is even: negatives do not flip cos. Sine does flip.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Cosine is even: negatives do not flip cos. Sine does flip.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
Cosine is even: negatives do not flip cos. Sine does flip.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
sin odd, cos even
360-A is a fourth-quadrant picture
12
cos even
sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A, and the rest as the book table. Meaning: even/odd plus CAST
- Definition
- sin(360-A)=-sin A, cos(360-A)=cos A. sin(-A)=-sin A (odd), cos(-A)=cos A (even). sin(270-A)=-cos A, and the rest as the book table. Meaning: even/odd plus CAST
- Example
- sin(-A)=-sin A
- Critical warning
- Cosine is even: negatives do not flip cos. Sine does flip.
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Exam Practice
Define
Q1. Define More Allied Forms (270, Negative).
Explain
Q2. Explain: NEGATIVE ANGLES.
Example
Q3. Give one clear example of More Allied Forms (270, Negative) and say why it fits.
Mistake
Q4. State one common mistake students make about More Allied Forms (270, Negative).
Exam note
Q5. Write a short exam note on More Allied Forms (270, Negative): definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.