MathematicsChapter 8: Inverse Trigonometric Functions and Their Graphs
Inverse Identities
sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
Learning objectives
- State the rule for inverse identities.
- Solve a inverse identities problem step by step.
- Check the result, restrictions, notation, and units.
Definition
sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
Where / symbols
- 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 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
- 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.
Worked example
- Worked example:
- arcsin(sin(5pi/6))=pi/6.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Inverse identities
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Inverse Identities
Sticky Notes · Formula and Worked Revision
Core Idea
- sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
Rule / Formula
- sin(arcsin x)=x on [-1,1], but arcsin(sin x)=x only inside the principal interval.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- arcsin(sin(5pi/6))=pi/6.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- 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.
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