Angles

Correct answer: B. ( \frac{3\pi}{4} )

( -\theta=-\frac{5\pi}{4} ), and ( -\frac{5\pi}{4}+2\pi=\frac{3\pi}{4} ). Add (2\pi) to a negative angle.

Correct answer: A. \(635^\circ\)

To find a principal angle, subtract multiples of \(360^\circ\) from the given angle. Since \(635^\circ-360^\circ=275^\circ\), the principal angle of \(635^\circ\) is \(275^\circ\). In contrast, \(725^\circ-720^\circ=5^\circ\), so it is not correct. Exam tip: Coterminal angles differ by an integral multiple of \(360^\circ\).

Correct answer: B. Second quadrant

Subtracting three complete revolutions, where \(2\pi=\frac{12\pi}{6}\), gives \(\frac{41\pi}{6}-\frac{36\pi}{6}=\frac{5\pi}{6}\). Now \(\frac{5\pi}{6}=150^\circ\), which lies between \(\frac{\pi}{2}\) and \(\pi\); hence its terminal side lies in the second quadrant. The fourth quadrant contains angles between \(\frac{3\pi}{2}\) and \(2\pi\). Exam tip: reduce a large angle by subtracting multiples of \(2\pi\) before identifying its quadrant.

Correct answer: D. ( \frac{9\pi}{5} )

( \theta-\pi=\frac{14\pi}{5}-\frac{5\pi}{5}=\frac{9\pi}{5} ), and it is in (0) to (2\pi). First subtract and then check the range.

Correct answer: D. Negative (y)-axis

(3\theta=\frac{15\pi}{6}=\frac{5\pi}{2}), and its principal angle is ( \frac{\pi}{2} ). Hence the terminal side lies on the positive (y)-axis.

Correct answer: A. (70^\circ)

( \frac{7\pi}{18}\times\frac{180^\circ}{\pi}=70^\circ ). Multiply by ( \frac{180^\circ}{\pi} ) to convert radians to degrees.

Correct answer: A. \(\theta+2n\pi,\; n\in\mathbb{Z}\)

One complete revolution measures \(2\pi\) radians. Hence, adding or subtracting \(2n\pi\) from \(\theta\) leaves its terminal arm unchanged. Adding \(n\pi\) generally gives the opposite arm. Exam tip: use \(2\pi\) for coterminal angles.

Correct answer: A. \( \frac{\pi}{4} \)

Coterminal angles have the same terminal side because they differ by an integral multiple of \(2\pi\). Here, \(\frac{9\pi}{4}-2\pi=\frac{9\pi}{4}-\frac{8\pi}{4}=\frac{\pi}{4}\). Therefore, the terminal side is coterminal with \(\frac{\pi}{4}\). Although \(\frac{\pi}{2}\) may seem close, it represents \(90^\circ\), whereas \(\frac{\pi}{4}=45^\circ\). Exam tip: reduce an angle to the interval \([0,2\pi)\) by adding or subtracting \(2\pi\) as needed.

Correct answer: B. (215^\circ)

In the third quadrant the principal angle is (180^\circ+35^\circ=215^\circ). Add or subtract the reference angle according to the quadrant.

Correct answer: C. ( \frac{17\pi}{9} )

In the fourth quadrant ( \theta=2\pi-\frac{\pi}{9}=\frac{17\pi}{9} ). A full revolution in radians is (2\pi).

Correct answer: D. No positive value

( \frac{7\pi}{9}=140^\circ ), which is greater than (90^\circ), so no positive complementary angle is possible. First make the units same.

Correct answer: C. (55^\circ)

( \frac{11\pi}{6}=330^\circ ), and the other angle is (330^\circ-275^\circ=55^\circ). First make the units of both angles the same.

Correct answer: A. First quadrant

Subtracting one full revolution, \(765^\circ-720^\circ=45^\circ\). Since \(45^\circ\) lies between \(0^\circ\) and \(90^\circ\), the terminal side is in the first quadrant. The positive \(y\)-axis would require an angle of \(90^\circ\), or a coterminal angle. Exam tip: reduce any angle by multiples of \(360^\circ\) before identifying its quadrant or axis.

Correct answer: C. 16 cm

The arc-length formula is \(s=r\theta\), where \(\theta\) must be in radians. Thus, \(r=\frac{s}{\theta}=\frac{12\pi}{3\pi/4}=12\pi\times\frac{4}{3\pi}=16\) cm. Therefore, option C is correct. An error such as choosing 12 cm can result from dividing incorrectly by \(\frac{3\pi}{4}\). Exam tip: In arc-length questions, first check that the angle is expressed in radians.

Correct answer: A. ( \frac{\pi}{6} )

( -\frac{13\pi}{6}+\frac{24\pi}{6}=\frac{11\pi}{6} ), and the reference angle is ( \frac{\pi}{6} ). First bring the negative angle into (0) to (2\pi).

Correct answer: A. ( \frac{\pi}{3} )

( \frac{4\pi}{3}+3\pi=\frac{13\pi}{3} ), and ( \frac{13\pi}{3}-4\pi=\frac{\pi}{3} ). Subtract multiples of (2\pi).

Correct answer: B. \(285^\circ\)

Coterminal angles differ by an integral multiple of \(360^\circ\). Since \(-75^\circ+360^\circ=285^\circ\), \(285^\circ\) is coterminal with \(-75^\circ\). The difference between \(255^\circ\) and \(-75^\circ\) is \(330^\circ\), which is not a multiple of \(360^\circ\). Exam tip: add or subtract \(360^\circ\) to check coterminal angles quickly.

Correct answer: C. 25

To convert degrees to radians, multiply by \(\frac{\pi}{180}\): \(250^\circ=250\times\frac{\pi}{180}=\frac{25\pi}{18}\). Hence, \(\frac{x\pi}{18}=\frac{25\pi}{18}\), so \(x=25\). The nearby option 24 is incorrect because it corresponds to \(240^\circ\). Exam tip: remember \(\theta^\circ=\frac{\theta\pi}{180}\) radians.

Correct answer: A. (207^\circ)

( \frac{23\pi}{20}\times\frac{180^\circ}{\pi}=207^\circ). Use (180^\circ\div20=9^\circ) and multiply.

Correct answer: B. (27) square cm

Sector area is ( \frac{1}{2}rs=\frac{1}{2}\times6\times9=27 ). Use this short formula when arc length is given.

Correct answer: A. They are coterminal

The difference ( \frac{29\pi}{12}-\frac{5\pi}{12}=2\pi ), so they are coterminal. In radians, coterminal angles differ by a multiple of (2\pi).

Correct answer: A. (67^\circ30')

The direct answer is 67°30′, so option A is correct. A supplementary angle is the angle that makes a straight angle of 180°. Therefore subtract the given angle from 180°: 180°00′−112°30′. Since 0 minutes cannot subtract 30 minutes, borrow 1 degree, which is 60 minutes. Then 179°60′−112°30′=67°30′. Option A, 67°30′, works exactly. Option B, 77°30′, would leave a total of 190°, not 180°. Option C, 57°30′, gives 170° when added to the given angle. Option D, 68°30′, gives 181°, so it is one degree too large. The word supplementary means a total of 180°, while complementary means a total of 90°; confusing these words is a common error. Memory cue: for a supplementary angle, use 180°−θ, and borrow 60 minutes when the minutes of the first angle are zero.

Correct answer: A. \(\frac{5\pi}{6}\)

Add one full turn, \(2\pi\), to the given angle: \(-\frac{7\pi}{6}+2\pi=\frac{5\pi}{6}\). This lies between 0 and \(2\pi\), so the answer is \(\frac{5\pi}{6}\).

Correct answer: B. \(50^\circ\)

Convert radians to degrees by multiplying by \(\frac{180^\circ}{\pi}\): \(x=\frac{5\pi}{18}\times\frac{180^\circ}{\pi}=50^\circ\).

Correct answer: B. \(100^\circ\)

Subtract three full turns: \(1180^\circ-3(360^\circ)=100^\circ\). This is in the stated interval, so the coterminal angle is \(100^\circ\).