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Forces A and B has a resultant force C with magnitude of 200N.
The magnitudes of A and B have the relation of 2||A||=3||B||. $\theta$ is the angle between A and C, and the angle between B and C is 2$\theta$. Determine the value of $\cos\theta$ and $\cos2\theta$. Then find ||A|| and ||B||.

General tips to get me started on this is appreciated.

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1 Answer 1

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Hint: Given the angles, the angle between $A$ and $B$ will be $3\theta$ (As $A,B,C$ must lie in a plane). We have $$\tag12|A|=3|B|$$ $$\tag2|C|=\sqrt{|A|^2+|B|^2+2|A||B|\cos3\theta}$$

If the angle between $A$ and $C$ is $\alpha$ and that between $A$ and $B$ is $\theta$, then $$\tag3\tan\alpha=\frac{|B|\sin\theta}{|A|+|B|\cos\theta}$$ Here, replace $\alpha$ with $\theta$ and $\theta$ with $3\theta$. $$\tag4|C|=200$$

Use $(1)$, $(3)$ and $(4)$ to substitute in $(2)$

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  • $\begingroup$ Hey thanks for the hint. However, I am to find $\cos\theta$ and $\cos2\theta$ prior to finding $\cos3\theta$, and then finally ||A|| ||B||. Is there a different way to approach this question in this order? $\endgroup$
    – Yuxie
    Apr 3, 2016 at 8:01
  • $\begingroup$ @Yuxie In $(3)$, you can write $\cos3\theta=4\cos^3\theta-3\cos\theta$, and using that find $\cos\theta$. Then, $\cos2\theta=2\cos^2\theta-1$. Finally, you can use these values and $(1)$ iin $(2)$ to get one of $|A|$ and $|B|$. Then, substitute back in $(1)$ to get the other. $\endgroup$
    – GoodDeeds
    Apr 3, 2016 at 8:27
  • $\begingroup$ Sorry for all these questions but, how would you use $\cos3\theta = 4\cos^3\theta-3\cos\theta$ in $(3)$ to find $\cos\theta$? $\endgroup$
    – Yuxie
    Apr 3, 2016 at 9:09

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