Two unknowns two equations with trig.

[holly2015]

Can anyone help me simplify this so i can find beta and gamma. (Theta is known).
200sin(beta)=190+80cos(theta)-240cos(gamma)
200cos(beta)=70+80sin(theta)+240sin(gamma)
thank you

 

[tkhunny]

#1 Simplify Your Life - Substitution will help a lot!!

200sin(beta)=190+80cos(theta)-240cos(gamma)
200cos(beta)=70+80sin(theta)+240sin(gamma)

Substitute A = sin(beta) and B = cos(beta)

200A=190+80cos(theta)-240cos(gamma)
200B=70+80sin(theta)+240sin(gamma)

Substitute C = sin(gamma) and D = cos(gamma)

200A=190+80cos(theta)-240D
200B=70+80sin(theta)+240C

Reduce - Divide by 10

20A=19+8cos(theta)-24D
20B=7+8sin(theta)+24C

Rewrite into a more standard form.

20A + 24D = 19 + 8cos(theta)
20B - 24C = 7 + 8sin(theta)

Observe additional relationships

A^2 + B^2 = 1
C^2 + D^2 = 1

That's enough information to solve. Can you do it? It's complicated enough, you may wish to stick with numeric solutions.