MathematicsChapter 8: Inverse Trigonometric Functions and Their Graphs
Compositions Outside Principal Ranges
First evaluate the trig value, then select the principal angle with that value.
Learning objectives
- State the rule for compositions outside principal ranges.
- Solve a compositions outside principal ranges problem step by step.
- Check the result, restrictions, notation, and units.
Definition
First evaluate the trig value, then select the principal angle with that value.
Where / symbols
- Compositions outside principal ranges
- First evaluate the trig value, then select the principal angle with that value.
CONCEPT AND RULE
First evaluate the trig value, then select the principal angle with that value.
METHOD
For compositions outside principal ranges, 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
arccos(cos(4pi/3))=2pi/3.
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 compositions outside principal ranges
- 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:
- arccos(cos(4pi/3))=2pi/3.
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
- Compositions outside principal ranges
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Compositions Outside Principal Ranges
Sticky Notes · Formula and Worked Revision
Core Idea
- First evaluate the trig value, then select the principal angle with that value.
Rule / Formula
- First evaluate the trig value, then select the principal angle with that value.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- arccos(cos(4pi/3))=2pi/3.
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
- First evaluate the trig value, then select the principal angle with that value.
- Worked pattern: arccos(cos(4pi/3))=2pi/3.
- Final check: domain, sign, notation, units, and reasonableness.
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