Cud u give me the formula of Law of Cosine on how to solve a triangle w/ 3 sides
given?
Im kind of confused with this problem:
a = 5, b = 6, c = 9
find all 3 angles A,B & C
$\displaystyle c^2 = a^2 + b^2 - 2ab \cdot cos( \gamma )$Originally Posted by ^_^Engineer_Adam^_^
Where $\displaystyle \gamma$ is the angle across from side c.
Thus $\displaystyle cos( \gamma ) = \frac{a^2 + b^2 - c^2}{2ab}$.
The other formulae simply permute the values of a, b, and c and need not be given.
So for example, to find the angle across from side c we have:
$\displaystyle cos( \gamma ) = \frac{5^2 + 6^2 - 9^2}{2\cdot 5 \cdot 6} = -\frac{1}{3}$
Thus $\displaystyle \gamma$ is second quadrant and $\displaystyle \gamma \approx 109.5^o $
To find the angle across from side a, use a = 6, b = 9, c = 5. etc.
-Dan
You can also find the angles by using the fact that.
$\displaystyle \frac{1}{2}ab\sin \gamma =A$
where, $\displaystyle A=\mbox{ area }$
And you can calculate area by Heron's Formula.
$\displaystyle A=\sqrt{s(s-a)(s-b)(s-c)}$
$\displaystyle s=\mbox{ semi-perimeter }$
Hello, Adam!
Cud u give me the formula of Law of Cosine on how to solve a triangle w/ 3 sides given?
Im kind of confused with this problem:
a = 5, b = 6, c = 9
find all 3 angles A,B & C
No, I'm confused . . .
You're familiar with the Law of Cosines
. . but you've never solved for an angle . . . ever?
Okay, just this once . . .
I assume you know that: .$\displaystyle a^2\:=\:b^2+c^2 - 2bc\cos A$
Rearrange the terms: .$\displaystyle 2bc\cos A\:=\:b^2 + c^2 - a^2$
Divide by $\displaystyle 2bc:\;\;\boxed{\cos A\:=\:\frac{b^2 + c^2 - a^2}{2bc}}$ . . . a formula for finding $\displaystyle \angle A.$
Similarly, we can derive formulas for the other two angles:
. . $\displaystyle \boxed{\cos B\:=\:\frac{a^2+c^2-b^2}{2ac}}\qquad\boxed{\cos C\:=\:\frac{a^2+b^2-c^2}{2ab}} $
You should memorize these or be able to derive them when needed.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Your problem has: .$\displaystyle a = 5,\lb = 6,\;c = 9$
We have: .$\displaystyle \cos A\:=\:\frac{6^2 + 9^2 - 5^2}{2(6)(9)}\:=\:\frac{92}{108}$
. . Therefore: .$\displaystyle A\:=\:\cos^{-1}\left(\frac{92}{108}\right)\:=\:31.5863381\quad \Rightarrow\quad \boxed{A\:\approx\:31.6^o}$
We have: .$\displaystyle \cos B\:=\:\frac{5^2 + 9^2 - 6^2}{2(5)(9)}\:=\:\frac{70}{90}$
. . Therefore: .$\displaystyle B\:=\:\cos^{-1}\left(\frac{7}{9}\right)\:=\:38.94244127\quad \Rightarrow\quad \boxed{B\:\approx\:38.9^o}$
We have: .$\displaystyle \cos C\:=\:\frac{5^2+6^2-9^2}{2(5)(6)}\:=\:\frac{-20}{60}$
. . Therefore: .$\displaystyle C\:=\:\cos^{-1}\left(-\frac{1}{3}\right)\:=\:109.4712206\quad \Rightarrow\quad \boxed{C\:\approx\:109.5^o}$
Check: .$\displaystyle A + B + C\:=\:31.6^o + 38.9^o + 109.5^o \:=\:180^o$ . . . Yay!
Yea ...
But the only confusing thing is that after solving the law of cosine to get the 1st angle A which is 32 degrees, i solve the angle using the law of sine... so sin32 / 5 = sin C / 9 and it gave the C an angle of 72.5 degrees...
How come?
Btw
Thanks topsquart, ThePerfectHacker and thanks again Soroban!!
Hello again, Adam!
After solving the Law of Cosine to get $\displaystyle A = 32^o$
i solved the angle using the Law of Sines.
So $\displaystyle \frac{\sin32}{5} = \frac{\sin C}{9}$ and it gave $\displaystyle C = 72.5^o.$
How come?
You fell for a very common "trap" in these problems.
Recall that an inverse sine can have two possible values.
. . For example: .$\displaystyle \sin^{-1}(0.5)\,=\,30^o$ or $\displaystyle 150^o$
And your calculator gives you only the smaller value.
It is up to you to determine which angle is appropriate.
You got: .$\displaystyle C = 72.5^o$, but $\displaystyle 107.5^o$ is also possible.
I've explained this to my students:
"The Law of Sines is much easier to use for determining angles.
But the Law of Sines (and your calculator) can lie to you.
. . (It says the angle is $\displaystyle 60^o$, but it's really $\displaystyle 120^o.)$
Hence, I recommend that you use the Law of Cosines to find angles.
. . (It doesn't lie.)"