if 1+sin²theta = 3sin theta cos theta ,prove that 2sin theta =cos theta
Answers
Step-by-step explanation:
If sinθ+2cosθ=1 , then what is a proof that 2sinθ−cosθ=2 ?
I’ll do the proof below, but let’s start with the counterexample.
Linear combinations of cosθ and sinθ can be viewed as a sum angle or difference angle formula, so wind up a combination rotation and scaling. The first equation works out to
cos(θ−something)=something
The right side wont be one, so there will be two θ values that satisfy sinθ+2cosθ=1. (We’re only considering angles between −180∘ and 180∘. )
One of them is θ=90∘, and in that case 2sinθ−cosθ=2✓
The other one is around θ=−36.7∘, which gives 2sinθ−cosθ=−2. Cue the deflated trombone sound effect. The statement to prove is not true.
Show, given sinθ+2cosθ=1 , that
2sinθ−cosθ=±2
Proof.
Let’s start with the 1,2,5–√ right triangle. Let t=arctan12 , right there in the first quadrant, t≈27∘, so
cost=25–√
sint=15–√
5–√cost=2
5–√sint=1
See where we’re going with this?
2cosθ+sinθ=1
5–√costcosθ+5–√sintsinθ=1
5–√cos(θ−t)=1
cos(θ−t)=1/5–√
We’ve seen this right triangle already. This tells us
sin(θ−t)=±2/5–√
That’s where the ambiguity pops up, in the form of a ±. Expanding with the difference angle formula for sine,
sinθcost−cosθsint=±2/5–√
(sinθ)(2/5–√)−(cosθ)(1/5–√)=±2/5–√
2sinθ−cosθ=±2✓