Calculus activities

Question

Create a chart of the rules for when to use the Law of Sines verses Law of Cosines, for a new student of Precalculus.

Answer 1

Law of Sines: SinA/a =SinB/b = SinC/c

where a is the side opposite angle A

b is the side opposite angle B

c is the side opposite angle C

If you know two sides and one opposite angle use the Law of Sines to find the other angle

you know a and b and A, use the Law of Sines to find angle B

know 2 sides and one angle opposite one of the sides use the Law of Sines to find the other opposite angle

or if you know 2 angles and one opposite side, use the Law of Sines to find the other opposite side

Law of Cosines: c^2 = a^2 +b^2 -2abCosC which is a generalized version of the Pythagorean Theorem including when C is not 90 degrees, when the triangle is not a right triangle.

IF you know 3 sides to find one of the angles, use the Law of Cosines

know a, b and c, then use the Law of Cosines to find angles A, B or C

You could then use the Law of Cosines to find all 3 angles

or switch back to the Law of Sines once you have 3 sides and 1 Angle to find the other two angles.

if all you know is 3 angles, there's no way to find any of the angles

if you know 2 sides and one angle, but the angle is not opposite either of the 2 known sides, such as you know b and c and angle A, then use the Law of Cosines to find side a: a^2 = b^c + c^2 -2bcCosA

in general if you know 2 sides and one angle opposite one of the sides: Use the Law of Sines to find the other angle opposite one of the known sides

If you know 2 angles and 1 side, you really know 3 angles, which includes the angle opposite the one known side, so use the Law of Sines. if you know 2 angles, such as A and B, then you know C = 180-A-B

Do a chart with what you know and what you can get from the known sides&angles, by the Law of Sines and Cosines.

Just a Idea

Maybe a row on top of the chart with what you know, 3A1S, 2A1S, 1A2S, ....

then left column of what you can get:

intersections either Law of Sines or Cosines, both or neither.

or maybe the left column just 3 entries, Law of Sines and Law of Cosines, and both

then the intersections as what you can get from using them given what you know.