Question: Two equation of two unknown angles (Hint: cos^2(theta_4)+sin^2(theta_4)=1)

[Amcw]

I have come to a stage in my work which involves finding two unknown angles theta3 and theta4.
This was suggested that it can be done using maths programmes like Matlab, however, I was stated that this could be done by hand and I was wondering if anyone knows how to do it.

The equations are:

3.0348+3.1783cos(theta3)+2.8790cos(theta4)=0
3.4272+3.1783sin(theta3)+2.8790sin(theta4)=0

How do you find theta3 and theta4?

 

[tkhunny]

You should use your hint and reduce your load.

 

[Steven G]

I have come to a stage in my work which involves finding two unknown angles theta3 and theta4.
This was suggested that it can be done using maths programmes like Matlab, however, I was stated that this could be done by hand and I was wondering if anyone knows how to do it.

The equations are:

3.0348+3.1783cos(theta3)+2.8790cos(theta4)=0
3.4272+3.1783sin(theta3)+2.8790sin(theta4)=0

How do you find theta3 and theta4?

Before Denis beats me to it, I would rewrite the equation to make it simpler
Let A = 3.0348, B= 3.4272, C= 3.1783 and D= 2.8790

So the 2 equations become:

A+Ccos(theta3)+Dcos(theta4)=0 and
B+Csin(theta3)+Dsin(theta4)=0

Now as already pointed out, use your hint. How do you think that you can use it??

 

[Amcw]

I've tried multiplying eq1 by cos(theta4) and eq2 by sin(theta4) then adding the two equations together and rearranging but that didn't seem to make it simpler.

I've also thought about rearranging the hint and seeing if that would help simplify anything but couldn't see how it would.

This is the first time I've come across a problem involving two unknown angles and I'm completely stuck. Any reading materials, simpler examples or how I should go about it would be appreciated. Time is a factor here.

 

[Deleted member 4993]

Before Denis beats me to it, I would rewrite the equation to make it simpler
Let A = 3.0348, B= 3.4272, C= 3.1783 and D= 2.8790

So the 2 equations become:

A+Ccos(theta3)+Dcos(theta4)=0 and
B+Csin(theta3)+Dsin(theta4)=0

Now as already pointed out, use your hint. How do you think that you can use it??

Following up on Jomo's hint (Denis's disciple):

A+Ccos(theta3)+Dcos(theta4)=0 and
B+Csin(theta3)+Dsin(theta4)=0

A+Ccos(theta3)+Dcos(theta4)=0 →

cos(Θ3) = - A/C - D/C cos(Θ4) .................. (1) and

B+Csin(theta3)+Dsin(theta4)=0 →

sin(Θ3) = - B/C - D/C sin(Θ4) .....................(2)

What do you get when you "square" both sides of equations 1 & 2

cos²(Θ3) = [A/C + D/C cos(Θ4)]² .................. (3)

sin²(Θ3) = [B/C + D/C sin(Θ4)]² .....................(4)

Now "add" (3) and (4) → what do you get?

 

[Amcw]

Thanks that really helps me progress to here but not sure what I need to do now, I just need to rearrange for theta4.

 

[Deleted member 4993]

Thanks that really helps me progress to here but not sure what I need to do now, I just need to rearrange for theta4.

What you have can be rewritten as:

P* cos(Φ) + Q * sin(Φ) = R

Divide both sides by √(P² + Q²)

Substitute:

cos(Ω) = P/√(P² + Q²)

sin(Ω) = Q/√(P² + Q²) and

cos(δ) = R/√(P² + Q²) [.... assuming|R/√(P² + Q²)| ≤ 1]

What do you get now?