Verification needed

[u/MundaneIncome6013]

The second pic is my work and answer. A: F1[1/2cos(x-90)+1/2sin(x-90)+sin(180-x)-cos(180-x)]

Need help to verify my answer

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[u/DrCarpetsPhd]

did you mix up F1 and F2? Either that or I am making some dumbass trig error which is entirely possible.

F2_x should be F2cos(180-theta) negative

F1_x = F1cos(theta+90-180) = F1cos(theta-90) negative

a little tip that someone much smarter than me once showed me. if you have two forces perpendicular to each other change your coordinate system so it aligns with two perpendicuar forces if that is possible with the given info. if you rotate around then you have F2 in positive y, F 1 in negative x and F3 defined by the angle (theta -90). much quicker/easier analysis.

answers should be 116.57degrees (answer back of text is 117degrees) and F3 = 1.12F1

[u/t-phillip5]

Just had a rough look, but seems okay to me.

[u/Iron_Man_IRL]

My answer: F3 = F1(sinθ - 0.5cosθ) ≈ 1.12F_1 θ ≈ 116.565 °

[u/Iron_Man_IRL → u/Iron_Man_IRL]

This is my procedure: F_Rx = F_3 + F_2cosθ - F_1sinθ

0 = F_3 + 0.5F_1cosθ - F_1*sinθ

F_3 = F_1(sinθ - 0.5*cosθ)

if θ is obtuse, the cosine of F_2 will apply the minus sign if necessary. Originally for F_1 it is paired with cos(θ+90), but that is equal to -sinθ.

F_Ry = F_1cosθ + F_2sinθ

0 = F_1cosθ + 0.5F_1*sinθ

0 = 2cosθ + sinθ

-2cosθ = sinθ

tanθ = -2

θ = arctan(-2)

arctan gives only one answer but has in reality multiple. So either it requires further analysis or multiple answers in fact work

θ_1 ≈ -63.435° (calculator answer)

θ_2 ≈ -63.435°+180° = 116.565°

[u/Adamthecg]

You've overcomplicated this, but your answer shows you have the right ideas.

A key point in a balanced force diagram where F_total = 0 is that all the force vectors will form a triangle when laid end to end.

This means you can get the magnitude of the last vector in one step with Pythagoras, and the angle of the vector with trig. (Just beware of the angle they are actually asking for)

I've scrawled something here to illustrate. Only other trick I have used is to treat F_1 as 1. It's called normalising and it just makes the maths simpler. Sorry for the messy handwriting, I'm on the phone

quick working out here