Prove the inscribed angle theorem using vectors. I fixed the dots A=(cos theta,sin theta), and B=(cos psi, sin psi), and I took another point C=(cos psi,sin psi) in the biggest arc AB.

ct1a2n4k

ct1a2n4k

Answered question

2022-09-23

I am trying to prove the inscribed angle theorem using vectors. I fixed the dots A = ( cos θ , sin θ ), and B = ( cos φ , sin φ ), and I took another point C = ( cos ψ , sin ψ ) in the biggest arc AB.
My idea was to calculate A C , B C | A B | | B C | , what according to my calculations is 1 + cos ( θ φ ) cos ( ψ θ ) cos ( φ ψ ) 2 1 + cos ( θ ψ ) cos ( φ ψ ) cos ( θ ψ ) cos ( φ ψ ) ..
My main difficulty here is the square root, which I can't get rid of. Does someone know how to proceed from here?Or maybe to solve the problem with vectors with a different approach?

Answer & Explanation

Kailey Santana

Kailey Santana

Beginner2022-09-24Added 12 answers

Step 1
You expanded your terms all the way. Take a few steps back:
A C , B C = ( cos θ cos ψ , sin θ sin ψ ) , ( cos ϕ cos ψ , sin ϕ sin ψ ) = ( cos θ cos ψ ) ( cos ϕ cos ψ ) + ( sin θ sin ψ ) ( sin ϕ sin ψ ) = 2 sin θ ψ 2 sin θ + ψ 2 ( 2 ) sin ϕ ψ 2 sin ϕ + ψ 2 + 2 cos θ + ψ 2 sin θ ψ 2 2 cos ϕ + ψ 2 sin ϕ ψ 2 = 4 sin θ ψ 2 sin ϕ ψ 2 ( sin θ + ψ 2 sin ϕ + ψ 2 + cos θ + ψ 2 cos ϕ + ψ 2 ) = 4 sin θ ψ 2 sin ϕ ψ 2 cos ( θ + ψ 2 ϕ + ψ 2 ) = 4 sin θ ψ 2 sin ϕ ψ 2 cos θ ϕ 2
| A C | 2 | B C | 2 = ( ( cos θ cos ψ ) 2 + ( sin θ sin ψ ) 2 ) ( ( cos ϕ cos ψ ) 2 + ( sin ϕ sin ψ ) 2 ) = ( 2 2 cos θ cos ψ 2 sin θ sin ψ ) ( 2 2 cos ϕ cos ψ 2 sin ϕ sin ψ ) = 4 ( 1 cos ( θ ψ ) ) ( 1 cos ( ϕ ψ ) ) = 4 2 sin 2 θ ψ 2 2 sin 2 ϕ ψ 2 = 16 sin 2 θ ψ 2 sin 2 ϕ ψ 2
Step 2
and hence cos A C B = A C , B C | A C | | B C | = cos θ ϕ 2 which means that the angle A C B does not depend on C and is equal to half the central angle:
A C B = 1 2 ( θ ϕ ) = 1 2 A O B .

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