MathematicsChapter 9: Solution of Trigonometric Equations
Quadratic Equations in One Trig Function
Substitute u=sin x or cos x, solve the quadratic, reject values outside [-1,1], then find angles.
Learning objectives
- State the rule for quadratic equations in one trig function.
- Solve a quadratic equations in one trig function problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Substitute u=sin x or cos x, solve the quadratic, reject values outside [-1,1], then find angles.
Mathematical form
u=sin x or cos x
x=1 or 12.
Where / symbols
- Quadratic equations in one trig function
- Substitute u=sin x or cos x, solve the quadratic, reject values outside [-1,1], then find angles.
CONCEPT AND RULE
Substitute u=sin x or cos x, solve the quadratic, reject values outside [-1,1], then find angles.
METHOD
For quadratic equations in one trig function, 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
2sin2x-3sin x+1=0 gives sin x=1 or 12.
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 quadratic equations in one trig function
- 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:
- 2sin2x-3sin x+1=0 gives sin x=1 or 12.
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
- Quadratic equations in one trig function
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Quadratic Equations in One Trig Function
Sticky Notes · Formula and Worked Revision
Core Idea
- Substitute u=sin x or cos x, solve the quadratic, reject values outside [-1,1], then find angles.
Rule / Formula
- Substitute u=sin x or cos x, solve the quadratic, reject values outside [-1,1], then find angles.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- 2sin2x-3sin x+1=0 gives sin x=1 or 12.
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
- Substitute u=sin x or cos x, solve the quadratic, reject values outside [-1,1], then find angles.
- Worked pattern: 2sin2x-3sin x+1=0 gives sin x=1 or 12.
- Final check: domain, sign, notation, units, and reasonableness.
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