Maths Extension 1 Year 11 - Module 3 - Lesson 7
Half Angle Formulas
Name
Date
Class
1. Lesson focus
Half-angle formulae. Half-angle signs depend on the quadrant of x/2; for 0<x<π the principal square roots for sin(x/2) and cos(x/2) are positive.
Success criteria
- Apply the lesson method accurately in a routine example.
- Show each algebraic or domain step and check the result.
- Explain why the method and restrictions fit the question.
2. Worked practice
2. Find cos(π/8) exactly.
3. Find cos(3π/8) exactly.
3. Challenge
Solve sin(x/2)=1/2 for 0≤x<2π.
4. Multiple choice
For 0≤x≤π, which is sin(x/2) in square-root form?
A. −√((1−cosx)/2)
B. √((1+cosx)/2)
C. √((1−cosx)/2)
D. 2sinxcosx
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. Find sin(π/8) exactly.
sin(π/8)=√((1−cos(π/4))/2)=√(2−√2)/2.
2. Find cos(π/8) exactly.
cos(π/8)=√((1+cos(π/4))/2)=√(2+√2)/2.
3. Find cos(3π/8) exactly.
cos(3π/8)=sin(π/8)=√(2−√2)/2.
Challenge. Solve sin(x/2)=1/2 for 0≤x<2π.
Let u=x/2∈[0,π). sin u=1/2 at π/6 and 5π/6; hence x=π/3,5π/3.
Multiple choice. For 0≤x≤π, which is sin(x/2) in square-root form?
Correct answer: C. Here x/2∈[0,π/2], where sine is non-negative. Therefore sin(x/2)=√((1−cos x)/2), using the positive root.
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