Question

if 1+sin²theta = 3sin theta cos theta ,prove that 2sin theta =cos theta​

Answer 1

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✓