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?

Question

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&#039;t get rid of. Does someone know how to proceed from here?Or maybe to solve the problem with vectors with a different approach?

Kailey Santana · Accepted Answer

Step 1You 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 2and 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  .

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.

Answered question

Answer & Explanation

New Questions in High school geometry