Problem 87 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (if possible) the complement and supplement of the angle. $$\frac{\pi}{3}$$

Step-by-Step Solution

Answer

The complement of \(\frac{\pi}{3}\) is \(\frac{\pi}{6}\) and its supplement is \(\frac{2\pi}{3}\).

Step 1: Finding the Complement

To find the complement of the angle, subtract the angle from \(\frac{\pi}{2}\). So, the complement of \(\frac{\pi}{3}\) is \(\frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6}\)

Step 2: Finding the Supplement

To find the supplement of the angle, subtract the angle from \(\pi\). So, the supplement of \(\frac{\pi}{3}\) is \(\pi - \frac{\pi}{3} = \frac{2\pi}{3}\)

Key Concepts

Complementary Angles

Complementary angles are two angles that add up to 90 degrees or \( \frac{\pi}{2} \) radians. For trigonometric calculations, this complementary relationship can be easily visualized as part of a right triangle. One angle completes the rest to make exactly 90 degrees.
When the angle is given in radians, just like \( \frac{\pi}{3} \), you find its complement by subtracting it from \( \frac{\pi}{2} \). Let's see how it works:

• Start with the total for complementary angles: \( \frac{\pi}{2} \)
• Subtract your angle: \( \frac{\pi}{2} - \frac{\pi}{3} \)
• This results in \( \frac{\pi}{6} \)

So, the complement of \( \frac{\pi}{3} \) is \( \frac{\pi}{6} \). Understanding this ensures smooth calculations in trigonometric problems.
It’s crucial to remember that complementary angles only exist if the original angle is less than \( \frac{\pi}{2} \). If not, it doesn’t have a complement in this context.

Supplementary Angles

Supplementary angles are pairs of angles that add together to make 180 degrees or \( \pi \) radians. This forms a straight line, making supplementary angles very important when working with linear geometry.
To find the supplement of an angle measured in radians, you subtract it from \( \pi \). Here’s how to calculate it for \( \frac{\pi}{3} \):

• Start with the total for supplementary angles: \( \pi \)
• Subtract your angle: \( \pi - \frac{\pi}{3} \)
• This gives you \( \frac{2\pi}{3} \)

Thus, the supplement of \( \frac{\pi}{3} \) is \( \frac{2\pi}{3} \).
This knowledge is fundamental, as it helps you visualize how different angles combine and interact. For example, in the context of a linear pair, knowing one angle automatically lets you find the other by using this concept.

Radians

Radians are a unit of angular measurement used in many areas of mathematics. Unlike degrees, which are based on dividing a circle into 360 equal parts, radians come from the radius of the circle.
One full circle is \( 2\pi \) radians, equivalent to 360 degrees. Hence, half a circle or a straight angle is \( \pi \) radians, which is 180 degrees:

• \( \pi \) radians = 180 degrees
• \( \frac{\pi}{2} \) radians = 90 degrees
• \( \frac{\pi}{3} \) radians ≈ 60 degrees

Understanding radians is key because they provide a more natural and powerful way to work with angles, especially in the fields of trigonometry, calculus, and physics. These units make calculations involving circular motion and periodic functions more straightforward and elegant.
If you're just starting with radians, try converting between radians and degrees until both become intuitive. This will enhance your ability to navigate through various math problems more adeptly.