Jaime biked 5 1/4 miles on Monday, 6 7/8 miles on Tuesday, and 8 1/2 miles on Wednesday. If he continues the pattern, how many m
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Answer:
<u>Given triangle:</u>
- ΔXYZ
- X(-1,2), Y(2,1), Z(-1, -2)
<u>D₀.₅(ΔXYZ) results in:</u>
- (x, y) → (0.5x, 0.5y)
- X'(-0.5, 1), Y'(1, 0.5), Z(-0.5, -1)
The cosine of an angle is the x-coordinate of the point where its terminal ray intersects the unit circle. So, we can draw a line at x=-1/2 and see where it intersects the unit circle. That will tell us possible values of θ/2.
We find that vertical line intersects the unit circle at points where the rays make an angle of ±120° with the positive x-axis. If you consider only positive angles, these angles are 120° = 2π/3 radians, or 240° = 4π/3 radians. Since these are values of θ/2, the corresponding values of θ are double these values.
a) The cosine values repeat every 2π, so the general form of the smallest angle will be
... θ = 2(2π/3 + 2kπ) = 4π/3 + 4kπ
b) Similarly, the values repeat for the larger angle every 2π, so the general form of that is
... θ = 2(4π/3 + 2kπ) = 8π/3 + 4kπ
c) Using these expressions with k=0, 1, 2, we get
... θ = {4π/3, 8π/3, 16π/3, 20π/3, 28π/3, 32π/3}
