Solving trigonometric equations: do I need to consider different cases in this example?I'm trying to solve this set of trigonometric equations: tan ⁡ ( δ + β 2 ) = c ⋅ tan ⁡ ( θ 2 ) b ⋅ tan ⁡ ( δ − β 2 ) = a cos ⁡ ( δ + β 2 ) cos ⁡ ( γ 2 ) = cos ⁡ ( θ 2 ) Where a,b,c and θ are some given (but random) real numbers (they could be 0 and π). The goal is to solve for the real values of γ , β and δ. I'm trying to solve these equations by combining some variables. Let M = δ + β 2 , N = δ − β 2 , P = γ 2 then it's easy to solve for M from equation 1. However, to solve for δ and β, we also need to figure out N. I'm worried that if I need to discuss the solutions by cases, like if θ = π (so that tan ⁡ θ 2 is undefined) or if a or b is equal to 0. Any suggestions on solving for this question? Is it possible to obtain a generalized solution? Thanks!!

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Solving trigonometric equations: do I need to consider different cases in this example?I&#039;m trying to solve this set of trigonometric equations:                                                        tan              ⁡                              (                                                      δ                    +                    β                                    2                                )                                                                  =              c              ⋅              tan              ⁡                              (                                  θ                  2                                )                                                                        b              ⋅              tan              ⁡                              (                                                      δ                    −                    β                                    2                                )                                                                  =              a                                                          cos              ⁡                              (                                                      δ                    +                    β                                    2                                )                            cos              ⁡                              (                                  γ                  2                                )                                                                  =              cos              ⁡                              (                                  θ                  2                                )                                                        Where a,b,c and   θ are some given (but random) real numbers (they could be   0 and   π). The goal is to solve for the real values of   γ  ,  β and   δ. I&#039;m trying to solve these equations by combining some variables. Let                    M        =                              δ            +            β                    2                ,        N        =                              δ            −            β                    2                ,        P        =                  γ          2                    then it&#039;s easy to solve for   M from equation   1. However, to solve for   δ and   β, we also need to figure out   N. I&#039;m worried that if I need to discuss the solutions by cases, like if   θ  =  π (so that   tan  ⁡      θ    2   is undefined) or if   a or   b is equal to   0. Any suggestions on solving for this question? Is it possible to obtain a generalized solution? Thanks!!

Kristen Garza · Accepted Answer

Look at:  M  +  N  =  δ  M  −  N  =  βYou can find   M  ,  N from your first two equations, and then calculate   δ  ,  β from those.For   γ, note that      tan    2    ⁡  (  α  )  +  1  =      1                  cos        2            ⁡      α      From this, you can write  cos  ⁡  α  =  ±      1                  tan        2            ⁡      α      +      1      (depending on the angle).Then this:  cos  ⁡      (                  δ        +        β            2        )    cos  ⁡      (          γ      2        )    =  cos  ⁡      (          θ      2        )  Becomes this:  cos  ⁡      (          γ      2        )    =  cos  ⁡      (          θ      2        )    ⋅            tan      2        ⁡    (                  δ        +        β            2        )    +    1

Solving trigonometric equations: do I need to consider different cases in this example? I'm trying to solve this set of trigonometric equations: tan((delta+beta)/(2)) = c* tan((theta)/(2)) b* tan((delta- beta)/(2)) = a cos ((delta+ beta)/(2)) cos((gamma)/(2)) = cos((theta)/(2))

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