Determining Angle with Vector Components

[srg]

Greetings women and men,

I have a problem in which I have to find an angle $\phi$.

http://srg.sdf.org/images/PF/StaticsHW.png

A horizontal force of $\vec{F}=400 lbs$ is placed on the structure at point A. Find angle $\phi$ to give the AB component of $\vec{F}$ a magnitude of 600 lbs.

To solve this, I drew a diagram
http://srg.sdf.org/images/PF/StaticsHW2.png

I used the Pythagorean theorem to find x: $$600^2=400^2+x^2 \rightarrow x=\sqrt{600^2-400^2} \therefore x=447.2$$

Then I used the law of sines to find $\phi$: $$\frac{447.2}{\sin{30}}=\frac{400}{\sin\phi} \rightarrow \frac{400\sin{30}}{447.2}=\sin\phi \rightarrow \arcsin{\frac{400\sin{30}}{447.2}}=\phi=26.6\deg$$

However, $\phi=26.6\deg$ is not the same as the answer in the back of my textbook. I'm not sure where I went wrong. The correct answer in the textbook is $\phi=38.3\deg$.

Thanks!

 

[jedishrfu]

Where's the right angle? The pythagorean theorem is predicated on using a right triangle.

 

[srg]

Where's the right angle? The pythagorean theorem is predicated on using a right triangle.

Ah! Duh! That's why. Law of cosines it is. Can't believe I overlooked that.

Thanks!

 

[phinds]

Uh ... do you understand the Pythagorean Theorem? What kind of triangles does it apply to?

EDIT: I see that jedishrfu beat me to it.