How does sin(65)/sin(37) become sin(65)/tan(37)?

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The discussion revolves around solving a 2D collision problem involving two pucks, where puck "a" collides with puck "b" at angles of 65° and 37° respectively. The user is trying to isolate the final velocity of puck "a" (Vfa) using trigonometric identities and algebra. The key equation provided in the textbook is Z = Y/[cos65 + (sin65/tan37)], prompting questions about how to derive this and the role of tan in the equation. Clarifications on trigonometric identities, specifically the relationship between sine, cosine, and tangent, are discussed to aid in solving for Vfa. The user expresses gratitude for the help received, indicating a better understanding of the trigonometric concepts involved.
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Homework Statement


This is collision in 2D problem. I already did most of the work except I'm stuck on the trigonometry and algebra for solving for the final velocity of object a.

I will put the problem here just in case someone wants to show me a better way for getting to the answer.

Collision between two pucks. Puck "a" has mass = 0.025kg Velocity of "a" = +5.5m/s

along x-axis "a" makes collision with puck "b" which has mass = 0.050kg and "b" starts at rest. Collision is not head on. So, after collision puck "a" flies apart from "b" at angle 65° and puck "b" flies off at angle 37°


Homework Equations



I am trying to solve for Vfa;

Ma*Voa = Ma*Vfa(cos65) + [Ma*Vfa(sin65)/(sin37)](cos37)

Ma = X
Voa = Y
Vfa = Z

XY = XZ(cos65) + [ XZ(sin65)/(sin37)](cos37) solve for Z

Hopefully that's better.

The solutions in textbook managed to solve for Vfa or Z like this:

Z = Y/[cos65 + (sin65/tan37)]

My question how to get Z by itself and where did (sin65/tan37) come from?
 
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foreverlost said:
My question how to get Z by itself
Factor out the Z from the right hand side.
and where did (sin65/tan37) come from?
What does tanθ equal? 1/tanθ?
 
Doc Al said:
Factor out the Z from the right hand side.

Okay What confused me was that I didn't know how to get rid of Z from inside the parenthesis.

What does tanθ equal? 1/tanθ?

I'm not sure could I just plug in any number to θ and figure it out? I just get different numbers.
 
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Doc Al said:
Factor out the Z from the right hand side.

What does tanθ equal? 1/tanθ?


tanθ = sin/cos

1/tanθ = cotθ = -sinθ/cosθ
 
foreverlost said:
tanθ = sin/cos
Right.

So 1/tanθ = 1/(sin/cos) = cosθ/sinθ

1/tanθ = cotθ = [STRIKE]-sinθ/cosθ[/STRIKE]
 
Doc Al said:
Right.

So 1/tanθ = 1/(sin/cos) = cosθ/sinθ

Thanks Doc Al. I always did have trouble with trigonometric identities. Fortunately this exact problem was on the exam, and I practically remembered every step without much thought lol.
 
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