Maths Extension 1 Year 11 - Module 3 - Lesson 8
The t-Formulas
Name
Date
Class
1. Lesson focus
The t-formulae. For t=tan(x/2), sinx=2t/(1+t²), cosx=(1−t²)/(1+t²), tanx=2t/(1−t²) where defined.
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. If t=2/3, find sin x and cos x.
3. If t=1 and 0≤x<2π, find x.
3. Challenge
Show that t=tan(x/2)=3/4 gives sinx=24/25 and cosx=7/25.
4. Multiple choice
For t=tan(x/2), which formula gives cos x?
A. 2t/(1−t²)
B. 2t/(1+t²)
C. (1+t²)/(1−t²)
D. (1−t²)/(1+t²)
Worked answer key
Compare each step with your working. Keep stated intervals, domains and restrictions in your final answer.
1. If t=1/2, find sin x, cos x and tan x.
sinx=4/5, cosx=3/5, tanx=4/3 by substitution in the t-formulae.
2. If t=2/3, find sin x and cos x.
sinx=(4/3)/(13/9)=12/13; cosx=(5/9)/(13/9)=5/13.
3. If t=1 and 0≤x<2π, find x.
tan(x/2)=1 with x/2∈[0,π); thus x/2=π/4 and x=π/2.
Challenge. Show that t=tan(x/2)=3/4 gives sinx=24/25 and cosx=7/25.
Substitute: sinx=(3/2)/(25/16)=24/25; cosx=(7/16)/(25/16)=7/25.
Multiple choice. For t=tan(x/2), which formula gives cos x?
Correct answer: D. The half-angle substitution converts the double-angle expressions into rational functions of t.
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