Proof of Trigonometric Equation with using Complex NumbersProve this identity without using complex numbers: P ( z , t ) = A cos ⁡ ( ω t − B z + θ 1 ) + D cos ⁡ ( ω t − B z + θ 2 ) = C cos ⁡ ( ω t − B z + θ t o t a l )Where C = ( A ) 2 + ( D ) 2 + 2 A D cos ⁡ ( θ 1 − θ 2 ) and θ t o t a l = tan − 1 ⁡ A sin ⁡ ( θ 1 ) + D sin ⁡ ( θ 2 ) A cos ⁡ ( θ 1 ) + D cos ⁡ ( θ 2 ) Using the trigonometric identities, I get add the two, but I cannot simplify them to get the correct answer. Any help would be greatly appreciated.

Question

Proof of Trigonometric Equation with using Complex NumbersProve this identity without using complex numbers:  P  (  z  ,  t  )  =  A  cos  ⁡  (      ω    t  −  B  z  +            θ        1    )  +  D  cos  ⁡  (      ω    t  −  B  z  +            θ        2    )  =  C  cos  ⁡  (      ω    t  −  B  z  +            θ              t      o      t      a      l        )Where   C  =      (    A          )      2        +    (    D          )      2        +    2    A    D    cos    ⁡    (                  θ            1        −                  θ            2        )   and            θ              t      o      t      a      l        =      tan          −      1        ⁡            A      sin      ⁡      (                        θ                1            )      +      D      sin      ⁡      (                        θ                2            )              A      cos      ⁡      (                        θ                1            )      +      D      cos      ⁡      (                        θ                2            )      Using the trigonometric identities, I get add the two, but I cannot simplify them to get the correct answer. Any help would be greatly appreciated.

hamputlnf · Accepted Answer

Here is a quick outline which you can complete for yourself1. For simplicity, write   x  =  ω  t  −  B  z2. Expand both sides using the compound angle formula for   cos  ⁡  (  A  +  B  )3. Compare coefficients of   cos  ⁡  x and of   sin  ⁡  x4. Divide these equations to get the required form for   tan  ⁡      θ          t      o      t      a      l      5. Square and add the equations to get the required form for       C    2

Proof of Trigonometric Equation with using Complex Numbers Prove this identity without using complex numbers: P(z, t) = A cos(omega t -Bz + θ_1) + D cos(omega t -Bz + θ_2) = C cos(omega t -Bz + θ_(total))

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