Question

Express sin(3theta) in terms of sin (theta)

Answer 1

First of all:-

3∅ can be called (2∅+∅)
so, sin(3∅) = sin(2∅ + ∅)
Open brackets on the RHS (right hand side)
= sin2∅ cos ∅ + cos2∅sin∅
This is simplified into;
= 2sin∅cos^2∅ + sin∅ (1- 2sin^2∅)
= sin∅ (2cos^2∅ + 1 - 2sin^2∅)
= sin∅ (2(1 - sin^2∅) +1-2sin^2∅)
= 3sin∅ - 4sin^3∅

Answer 2

3∅ can be rewritten as (2∅+∅)
sin(3∅) = sin(2∅ + ∅)
Opening brackets on the right hand side;
= sin2∅ cos ∅ + cos2∅sin∅
This simplifies to;
= 2sin∅cos^2∅ + sin∅ (1- 2sin^2∅)
= sin∅ (2cos^2∅ + 1 - 2sin^2∅)
= sin∅ (2(1 - sin^2∅) +1-2sin^2∅)
= 3sin∅ - 4sin^3∅