missing angles and sides?

Question

given:

gamma=40
a=3
b=4

Have to find:

Beta , alpha, c

please help me and explain this to me step by step, thanks my book sucks!!

Answer 1

OK--assuming alpha is opposite a, beta is opposite b, and gamma opposite c, it sounds like you just need to solve the triangle. You can do it with this information. Here's how:

Use law of cosines to find the missing side, so we can do the rest with law of sines and triangle properties. Why? Law of sines is just a ratio, and the math is easier than law of cosines.

So, to get side c, I plug it into law of cosines, using the form a^2 = b^2 + c^2 - 2ab cos(gamma). Substituting values, I get ~3.37 for c.

I then use law of sines to find a missing angle. I'll pick alpha. I use the form of law of sines a/cos(alpha) = c/cos(gamma). Substituting values, I get a alpha = 47 degrees.

Knowing 2 of the angles, and that the sum of the internal angles of a triangle add up to 180, beta = 180 - alpha - gamma, or beta = 93 degrees.

In general, you want to do the hardest work the least amount of times. So try to use the 180 property first, if that fails use law of sines, and if that fails go to law of cosines.

Finally, remember that if you are given 3 angles or 2 angles and an external side, you can't solve the triangle (no single unique solution).

Hope this helps.

Answer 2

If you are familiar with Cartesian coordinates and trig functions, let the angle gamma be at the origin, with side b along the positive X axis.

Then the coordinates for the other end of side a are given by:

x = a cos 40
y = a sin 40

The length of side c is then the magnitude of the vector from b to a or:

c = sqrt((b - x)^2 + y^2)

Since you have c and sin(gamma), you can compute their ratio. The law of sines gives you the sines of alpha and beta:

http://en.wikipedia.org/wiki/Law_of_sines

Given their sines, you can find the corresponding angles.

P.S. No matter how bad your book might be, your saying it "sucks" does you little good, and causes at least some of us to think less of you. Is that the result you intended?