91Ó°ÊÓ

Understanding angle measurement is fundamental in geometry and trigonometry. Angles can be measured in degrees or radians, with one complete revolution around a point being 360 degrees or 2\( \pi \) radians. Most trigonometric problems use radians. To convert between degrees and radians, remember that 180 degrees is equal to \( \pi \) radians.

Important angles to remember:

• \(90\degree = \frac{\pi}{2} \)
• \(180\degree = \pi \)

This knowledge is crucial when working with complementary and supplementary angles.

Trigonometry deals with the relationships between the angles and sides of triangles. Specifically, for our exercise, we need to understand the concepts of complementary and supplementary angles.

Complementary angles are two angles whose measures add up to \( \frac{\pi}{2} \) radians (or 90 degrees). For any angle \( \theta \), its complement is \( \frac{\pi}{2} - \theta \).

Supplementary angles are two angles whose measures add up to \( \pi \) radians (or 180 degrees). For any angle \( \theta \), its supplement is \( \pi - \theta \).

These concepts help solve various trigonometric problems, including finding specific angle measures and their related properties.

Using step-by-step solutions can make complex problems simpler and more manageable. For example, to find the complementary and supplementary angles of \( \frac{\pi}{3} \) radians:

Step 1: Define the complement. The complement of an angle \( \theta \) is \( \frac{\pi}{2} - \theta \).

Step 2: Define the supplement. The supplement of an angle \( \theta \) is \( \pi - \theta \).

Step 3: Calculate the complement. For \( \frac{\pi}{3} \), subtract it from \( \frac{\pi}{2} \):

\[ \frac{\pi}{2} - \frac{\pi}{3} = \frac{3\pi}{6} - \frac{2\pi}{6} = \frac{\pi}{6} \]

Step 4: Calculate the supplement. For \( \frac{\pi}{3} \), subtract it from \( \pi \):

\[\pi - \frac{\pi}{3} = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]
Following these steps will allow you to find complements and supplements for any angle with ease.