A jar contains 8 red marbles, 10 blue ones, and Red Blue Yellow 2 yellow ones. One marble is chosen at random. The color is reco
rded in the table, and then it is 14 returned to the jar. This is repeated 40 times. 19. Communicate Mathematical Ideas Use proportional reasoning to explain how you know that for each color, the theoretical and experimental probabilities are not the same.
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Answer: see proof below
<u>Step-by-step explanation:</u>
Given: A + B + C = π → C = π - (A + B)
→ sin C = sin(π - (A + B)) cos C = sin(π - (A + B))
→ sin C = sin (A + B) cos C = - cos(A + B)
Use the following Sum to Product Identity:
sin A + sin B = 2 cos[(A + B)/2] · sin [(A - B)/2]
cos A + cos B = 2 cos[(A + B)/2] · cos [(A - B)/2]
Use the following Double Angle Identity:
sin 2A = 2 sin A · cos A
<u>Proof LHS → RHS</u>
LHS: (sin 2A + sin 2B) + sin 2C
LHS = RHS: 4 cos A · cos B · sin C = 4 cos A · cos B · sin C
