Simultaneous trigonometric equations

[John O' Meara]

I am looking for help in solving a pair of simultaneous equations. I have not come across any maths book that solves trigonometric ones. I was wondering if I could get a step by step solution. Thanking you in advance for your time:

5.4=10cos(x) + 13.41cos(y) ....(i)
0=10sin(x) + 13.41sin(y) .....(ii)

 

[dextercioby]

Express "cos y" in terms of "cos x" using the II-nd equation and the relationship between the "sin" and the "cos".

Daniel.

 

[M98Ranger]

Hi there,

Solving simultaneous trigonometric equations can be challenging, but with the right approach, it can be done. I would be happy to help you with the pair of equations you have provided.

First, let's rearrange the equations to make them easier to work with:

(i) 10cos(x) + 13.41cos(y) = 5.4
(ii) 10sin(x) + 13.41sin(y) = 0

Next, we can use the Pythagorean identity (sin^2(x) + cos^2(x) = 1) to eliminate one of the variables. Let's square both sides of equation (ii) and substitute the value of sin^2(x) with (1-cos^2(x)):

100cos^2(x) + 267.5881sin^2(y) = 0
100cos^2(x) + 267.5881(1-cos^2(y)) = 0
100cos^2(x) + 267.5881 - 267.5881cos^2(y) = 0
100cos^2(x) - 167.5881cos^2(y) = -267.5881

Now, we can substitute this value into equation (i) and solve for cos(x):

10cos(x) + 13.41cos(y) = 5.4
10cos(x) + 13.41(1 - sin^2(y)) = 5.4
10cos(x) + 13.41 - 13.41sin^2(y) = 5.4
10cos(x) + 13.41 - 13.41(1 - cos^2(y)) = 5.4
10cos(x) - 13.41cos^2(y) = -8.01
10cos(x) = 13.41cos^2(y) - 8.01
cos(x) = (13.41cos^2(y) - 8.01)/10
cos(x) = (13.41cos(y) - 8.01)/10cos(y)

Now, we can substitute this value into equation (ii) and solve for sin(y):

10sin(x) + 13.41sin(y) = 0
10(1 - cos^2(y)) + 13.41