Answer:
A = 18.2°
B = 128.7°
C = 33.1°
Step-by-step explanation:
You want the measures of the angles in ∆ABC with a=4, b=10, and c=7.
Law of cosines
The law of cosines is best suited for finding the measure of an angle in a triangle when the side lengths are known. It can be written as ...
c² = a² +b² -2ab·cos(C)
Solving for the angle C, we have ...
[tex]C=\cos^{-1}\left(\dfrac{a^2+b^2-c^2}{2ab}\right)[/tex]
Application
Often, the law of cosines can be used to find the measure of one of the angles, then another can be found using the law of sines. The final angle can be found from the other two using the angle sum theorem.
When feasible, it can work well to write a function to compute the angle from a, b, c. This is what we have done in the attachment. The angle it finds is the one opposite side 'c', which is the last of the three function arguments. (The other two arguments may be in either order.)
If the law of sines is to be used, it is convenient to use the law of cosines to find the largest angle first. That avoids the ambiguity that might otherwise arise from the use of the law of sines. In any event, there is no ambiguity involved if the law of cosines is used to find every angle.
The angles are ...
[tex]A=\cos^{-1}\left(\dfrac{b^2+c^2-a^2}{2bc}\right)=\cos^{-1}\left(\dfrac{10^2+7^2-4^2}{2\cdot10\cdot7}\right)\\\\\\A=\cos^{-1}\left(\dfrac{133}{140}\right)\approx18.2^\circ\\\\\\B=\cos^{-1}\left(\dfrac{a^2+c^2-b^2}{2ac}\right)=\cos^{-1}(-0.625)\approx128.7^\circ\\\\\\C=\cos^{-1}(0.8375)\approx 33.1^\circ[/tex]
The angles area A = 18.2°, B = 128.7°, C = 33.1°.
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Additional comment
In the formula shown in the attachment, the factor 180/pi causes the angle to be converted from radians to degrees. As with most spreadsheets and scientific calculators, the default angle mode for the trig functions is radians.
