An angle \theta in standard position measures -750^\circ. Let \phi be the coterminal angle… — ACT

An angle $\theta$ in standard position measures $-750^\circ$ . Let $\phi$ be the coterminal angle of $\theta$ such that $0 \le \phi < 2\pi$ radians. If $\phi = \frac{a\pi}{b}$ , where $a$ and $b$ are positive integers with no common factors, what is the value of $a + b$ ?

Answer: 17

Answer

To find the coterminal angle

\phi

in the range

[0, 2\pi)

radians, we first determine the coterminal angle in degrees by adding multiples of

360^\circ

. Adding

3 \times 360^\circ = 1080^\circ

-750^\circ

results in

330^\circ

. We then convert

330^\circ

to radians by multiplying by

\frac{\pi}{180^\circ}

, which simplifies to

\frac{11\pi}{6}

. Since

11

and

6

are positive integers with no common factors,

a = 11

and

b = 6

. The sum

a + b

11 + 6 = 17

Step-by-Step Solution

Find the positive coterminal angle of

-750^\circ

within one full rotation.

330^\circ

Adding

1080^\circ

(three full rotations of

360^\circ

) to

-750^\circ

brings the angle within the standard range of

[0^\circ, 360^\circ)

Convert the angle from degrees to radians.

\frac{11\pi}{6}

radians

Multiplying the degree measure by

\frac{\pi}{180^\circ}

and simplifying the fraction converts it to radians.

Identify the values of

a

and

b

and compute their sum.

17

Comparing

\frac{11\pi}{6}

\frac{a\pi}{b}

shows

a = 11

and

b = 6

, which share no common factors. Their sum is

11 + 6 = 17

Key Concept

Finding coterminal angles and converting angle measures between degrees and radians.

Estimated Time:1m 30s

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