In the xy-plane, an angle \theta in standard position has its terminal ray intersecting th… — SAT

In the $xy$ -plane, an angle $\theta$ in standard position has its terminal ray intersecting the unit circle at the point $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$ . The angle is rotated counterclockwise by $15^\circ$ , and then its measure is doubled. The final resulting angle is coterminal with an angle of $\frac{a\pi}{b}$ radians, where $0 \le \frac{a\pi}{b} < 2\pi$ , and $a$ and $b$ are positive integers with no common factors. What is the value of $a + b$ ?

Answer: 5

Answer

The correct answer is 5. Starting with the point

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

on the unit circle, the angle in standard position is

225^\circ

(or

\frac{5\pi}{4}

radians). Adding

15^\circ

for the counterclockwise rotation yields

240^\circ

(or

\frac{4\pi}{3}

radians). Doubling this measure gives

480^\circ

(or

\frac{8\pi}{3}

radians). Finding the coterminal angle within

[0, 2\pi)

yields

120^\circ

(or

\frac{2\pi}{3}

radians). In this form,

a=2

and

b=3

, which are positive integers with no common factors, so

a+b = 5

Step-by-Step Solution

Determine the initial angle of the terminal ray intersecting the unit circle at

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

The initial angle

\theta

225^\circ

(or

\frac{5\pi}{4}

radians).

Since both the

x

- and

y

-coordinates are negative and equal, the angle lies in the third quadrant and forms a

45^\circ

reference angle with the negative

x

-axis, which corresponds to

225^\circ

Apply a counterclockwise rotation of

15^\circ

to the initial angle.

The new angle is

240^\circ

(or

\frac{4\pi}{3}

radians).

Counterclockwise rotations correspond to adding positive angle measures:

225^\circ + 15^\circ = 240^\circ

Double the measure of the rotated angle.

The doubled angle is

480^\circ

(or

\frac{8\pi}{3}

radians).

The prompt specifies that the angle's measure is doubled after the rotation:

240^\circ \times 2 = 480^\circ

Find the coterminal angle within the standard interval

[0, 2\pi)

radians (or

[0^\circ, 360^\circ)

The coterminal angle is

120^\circ

(or

\frac{2\pi}{3}

radians).

To bring the angle back into the interval

[0^\circ, 360^\circ)

, subtract

360^\circ

(one full rotation):

480^\circ - 360^\circ = 120^\circ

. In radians,

120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3}

radians.

Identify

a

and

b

and calculate their sum.

a = 2

b = 3

, and

a + b = 5

The coterminal angle is expressed as

\frac{a\pi}{b}

in simplest form, so

a=2

and

b=3

, which are positive integers with no common factors.

Key Concept

Converting between radians and degrees, finding coterminal angles, and performing angle transformations.

Estimated Time:3m 0s

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