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Difficulty: HardUnit Circle and Angle Measures (Radians and Degrees)

An angle θ\theta in standard position is rotated counterclockwise by 225∘225^\circ. The terminal ray of the resulting angle lies in the fourth quadrant along the line y=−x3y = -x\sqrt{3}. If the radian measure of the smallest positive angle θ\theta is written in simplest form as aπb\frac{a\pi}{b}, where aa and bb are positive integers, what is the value of a+ba + b?

Answer: 17

Answer

The value of a+ba + b is 1717.
The resulting angle α\alpha lies along the line y=−x3y = -x\sqrt{3} in the fourth quadrant, meaning its measure is 300∘300^\circ plus any multiple of 360∘360^\circ. Subtracting the counterclockwise rotation of 225∘225^\circ gives the original angle θ=75∘\theta = 75^\circ (for the smallest positive angle). Converting 75∘75^\circ to radians by multiplying by π180∘\frac{\pi}{180^\circ} yields 5π12\frac{5\pi}{12}. Since the fraction is in simplest form, a=5a = 5 and b=12b = 12, and their sum is 1717.

Step-by-Step Solution

1
Find the angle of the terminal ray after rotation from its equation and quadrant
The terminal ray after rotation is at an angle of 300∘300^\circ (or 5π3\frac{5\pi}{3} radians)
The line y=−x3y = -x\sqrt{3} has a slope of −3-\sqrt{3}, so the angle α\alpha in Quadrant IV satisfies tan⁡α=−3\tan\alpha = -\sqrt{3}, which means α=300∘\alpha = 300^\circ.
2
Set up and solve the equation for the original angle θ\theta before the counterclockwise rotation of 225∘225^\circ
θ=75∘+360∘k\theta = 75^\circ + 360^\circ k
Since the angle was rotated counterclockwise by 225∘225^\circ to reach the final position of 300∘300^\circ, we have θ+225∘=300∘+360∘k\theta + 225^\circ = 300^\circ + 360^\circ k.
3
Determine the smallest positive value of θ\theta
θ=75∘\theta = 75^\circ
Setting k=0k = 0 gives the smallest positive angle of 75∘75^\circ.
4
Convert the angle θ\theta from degrees to radians
θ=5π12\theta = \frac{5\pi}{12} radians
To convert degrees to radians, multiply by π180∘\frac{\pi}{180^\circ}, yielding 75π180=5π12\frac{75\pi}{180} = \frac{5\pi}{12}.
5
Calculate the sum of the numerator and denominator of the simplified radian fraction
a+b=17a + b = 17
The fraction 512\frac{5}{12} is in simplest form, so a=5a = 5 and b=12b = 12. The sum is 5+12=175 + 12 = 17.

Key Concept

Converting degree measures to radian measures and finding coterminal angles on the unit circle
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Topics

Unit Circle and Angle Measures (Radians and Degrees)

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