MathematicsChapter 9: Solution of Trigonometric Equations
Equations With Shifted Angles
Treat the whole shifted angle as one variable, solve, then subtract the shift.
Learning objectives
- State the rule for equations with shifted angles.
- Solve a equations with shifted angles problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Treat the whole shifted angle as one variable, solve, then subtract the shift.
Formula
x=pi/3+2npi.
Where / symbols
- Equations with shifted angles
- Treat the whole shifted angle as one variable, solve, then subtract the shift.
CONCEPT AND RULE
Treat the whole shifted angle as one variable, solve, then subtract the shift.
METHOD
For equations with shifted angles, 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
cos(x-pi/3)=1 gives x=pi/3+2npi.
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 equations with shifted angles
- 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:
- cos(x-pi/3)=1 gives x=pi/3+2npi.
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
- Equations with shifted angles
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Equations With Shifted Angles
Sticky Notes · Formula and Worked Revision
Core Idea
- Treat the whole shifted angle as one variable, solve, then subtract the shift.
Rule / Formula
- Treat the whole shifted angle as one variable, solve, then subtract the shift.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- cos(x-pi/3)=1 gives x=pi/3+2npi.
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
- Treat the whole shifted angle as one variable, solve, then subtract the shift.
- Worked pattern: cos(x-pi/3)=1 gives x=pi/3+2npi.
- Final check: domain, sign, notation, units, and reasonableness.
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