Question

Trig Identity in Mechanics Question Problem

I need help solving this problem. It is a mechanics equilibrium problem. The book gives the below equations, which are used to derive the answer below. However solving the equations requires a trig identity which no one knows how to do, including my lecturers. The book does not offer any solutions as to how it is done, but claims that it derives the answer from the said equations below.

->+ ∑F(x)=C*cos(30)-275*cos(theta)=0
Up+ ∑F(y)=C*sin(30)+275*sin(theta)-300=0
Answers (theta)=40.9 C=240

Answer 1

Just so you know, I am a university student and no one has been able to help me so far. Including my stupid statics teacher

Answer 2

did you try converting degree's to radians. That might help.

Answer 3

Yea I tried that, still end up needing some relationship between cos and sin...

Answer 4

If you multiply the first equation by sin(30) and the second equation by cos(30), then you can take the first equation away from the second. Then you can use the addition formulae to put the parts with theta in back into brackets and then it's just a matter of using arcsin and rearranging.

Answer 5

Is there any chance you could show me what you mean exactly?

Answer 6

I still end up with sin(theta)-cos(theta)

Answer 7

there is no trig identity that I am aware of that can solve that relation. I have my trig identity book in front of me. Maybe I am missing something

Answer 8

The first equation becomes \[Ccos(30)\sin(30) - 275\cos \theta \sin(30) = 0\] and the second becomes \[Csin(30)\cos(30) + 275\sin \theta \cos(30) - 300\cos(30) = 0 \]
So subtracting the first from the second, the first terms cancel so we get \[275(\sin \theta \cos(30) + \cos \theta \sin(30)) = 300\cos(30).\]

Answer 9

If I may ask what comes next? In your opinion, this is how far I got myself

Answer 10

Then using the addition formula we get \[275(\sin(\theta + 30)) = 150\sqrt{3}\]

Answer 11

I don't understand how you just removed the cos(theta). Using surds for the angles and all that I understand, but not the removal of cos(theta)

Answer 12

The addition formula says \[\sin(\theta + \alpha) = \sin \theta \cos \alpha + \sin \alpha \cos \theta\]

Answer 13

so in my case, theta =theta and alpha = 30?

Answer 14

Yes :)

Answer 15

may I just ask, what happens if it was sin(theta)*cos(30)+cos(theta)*sin(45)?

Answer 16

What happens with the different angles? do you add?

Answer 17

Then the addition formulae don't work anymore!

Answer 18

right, only same angles

Answer 19

Yes.

Answer 20

Thank you very much for that, you have solved a dilemma in my life that has been bugging me and a lot of students for a very long time :)

Answer 21

You're welcome :)

Answer 22

I have a question on this problem, when I solve it through I dont get the answer for theta of 40.9 degrees. Is there a step I am missing?