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Inverse Identities
sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
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Inverse Identities
What is it?
sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
In simple words
- sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
- Worked pattern: arcsin(sin(5pi/6))=pi/6.
- Final check: domain, sign, notation, units, and reasonableness.
Important terms
Inverse identities: sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
What you should learn
- State the rule for inverse identities.
- Solve a inverse identities problem step by step.
- Check the result, restrictions, notation, and units.
Recommended for Class 12
Full school-level explanation
sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
CONCEPT AND RULE
sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
METHOD
For inverse identities, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
arcsin(sin(5pi/6))=pi/6.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve inverse identities
How it works
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Remember
- Inverse identities
- rule
- worked example
- worked example
How the ideas connect
Worked example: arcsin(sin(5pi/6))=pi/6.
Deeper understanding
Go beyond memorising Inverse Identities. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked example: arcsin(sin(5pi/6))=pi/6.
Common mistakes
Exam focus
Core Idea
Rule / Formula
Method
Worked Example
Exam focus
Quick revision
- Inverse identities
- rule
- worked example
- Inverse identities: sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
- CONCEPT AND RULE: sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
- METHOD: For inverse identities, identify the given information and required result. Select the matching defi
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
For long answers on Inverse Identities: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- Inverse identities
- rule
- worked example
- Inverse identities: sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
Important formulas
- Inverse identities
- rule
- worked example
Common exam traps
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- A matching-looking formula is not valid until its conditions and domain have been checked.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
- Mistake
A matching-looking formula is not valid until its conditions and domain have been checked.
- Mistake
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Inverse identities
rule
worked example
Inverse identities: sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
- Definition
- sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
- Critical warning
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all d
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Exam Practice
Define
Q1. Define Inverse Identities.
Explain
Q2. Explain: CONCEPT AND RULE.
Example
Q3. Give one clear example of Inverse Identities and say why it fits.
Mistake
Q4. State one common mistake students make about Inverse Identities.
Exam note
Q5. Write a short exam note on Inverse Identities: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.