Question

refresher course math 1050 and 1060 practice problems set e spring 2008

1. convert to radian measure: a) \\(30^{\circ}\\)

b) \\(150^{\circ}\\)

1. convert to degree measure: a) \\(\frac{3\pi}{2}\\)

b) \\(-\frac{7\pi}{6}\\)

1. find two coterminal angles (one positive and one negative) for \\(\theta = 390^{\circ}\\).

1. find the complement and supplement of \\(\theta = 72^{\circ}\\).

1. find

a) the complement of \\(\theta = \frac{\pi}{12}\\).
b) the supplement of \\(\theta = \frac{5\pi}{6}\\).
c) a coterminal angle to \\(\theta = \frac{17\pi}{6}\\).

Explanation:

🆕 New Concept Discovered: Angle Relationships in Radians
Finding complements, supplements, and coterminal angles using radian measure.

Step 1: Find the complement of \(\theta = \frac{\pi}{12}\)

Two angles are complementary if their sum is \(90^\circ\), which is \(\frac{\pi}{2}\) radians. To find the complement, subtract the given angle from \(\frac{\pi}{2}\):

$$ \text{Complement} = \frac{\pi}{2} - \frac{\pi}{12} $$

Find a common denominator, which is \(12\):

$$ \frac{\pi}{2} - \frac{\pi}{12} = \frac{6\pi}{12} - \frac{\pi}{12} = \frac{5\pi}{12} $$

---

Step 2: Find the supplement of \(\theta = \frac{5\pi}{6}\)

Two angles are supplementary if their sum is \(180^\circ\), which is \(\pi\) radians. To find the supplement, subtract the given angle from \(\pi\):

$$ \text{Supplement} = \pi - \frac{5\pi}{6} $$

Find a common denominator, which is \(6\):

$$ \pi - \frac{5\pi}{6} = \frac{6\pi}{6} - \frac{5\pi}{6} = \frac{\pi}{6} $$

---

Step 3: Find a coterminal angle to \(\theta = \frac{17\pi}{6}\)

Coterminal angles share the same terminal side. We find them by adding or subtracting full rotations of \(360^\circ\), which is \(2\pi\) radians.

Subtracting one full rotation (\(2\pi\)):

$$ \frac{17\pi}{6} - 2\pi = \frac{17\pi}{6} - \frac{12\pi}{6} = \frac{5\pi}{6} $$

Alternatively, adding one full rotation (\(2\pi\)):

$$ \frac{17\pi}{6} + 2\pi = \frac{17\pi}{6} + \frac{12\pi}{6} = \frac{29\pi}{6} $$

Answer:

• a) Complement of \(\theta = \frac{\pi}{12}\): \(\frac{5\pi}{12}\)
• b) Supplement of \(\theta = \frac{5\pi}{6}\): \(\frac{\pi}{6}\)
• c) Coterminal angle to \(\theta = \frac{17\pi}{6}\): \(\frac{5\pi}{6}\) (or \(\frac{29\pi}{6}\))