Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Find (a) the complement and (b) the supplement of each angle. Do not use a calculator. $$\frac{\pi}{3}$$

Step-by-Step Solution

Answer

Complement: \( \frac{\pi}{6} \), Supplement: \( \frac{2\pi}{3} \).

Step 1: Understand Complementary Angles

Complementary angles are two angles whose sum is 90 degrees or \( \frac{\pi}{2} \) radians. So, if an angle is expressed in radians and we need to find its complement, we subtract the angle from \( \frac{\pi}{2} \).

Step 2: Calculate the Complement of \( \frac{\pi}{3} \)

To find the complement of \( \frac{\pi}{3} \), subtract \( \frac{\pi}{3} \) from \( \frac{\pi}{2} \): \[ \frac{\pi}{2} - \frac{\pi}{3} \]Convert \( \frac{\pi}{2} \) and \( \frac{\pi}{3} \) to have a common denominator:\[ \frac{3\pi}{6} - \frac{2\pi}{6} = \frac{\pi}{6} \]Thus, the complement of \( \frac{\pi}{3} \) is \( \frac{\pi}{6} \).

Step 3: Understand Supplementary Angles

Supplementary angles are two angles whose sum is 180 degrees or \( \pi \) radians. Thus, to find the supplement of an angle in radians, subtract the angle from \( \pi \).

Step 4: Calculate the Supplement of \( \frac{\pi}{3} \)

To find the supplement of \( \frac{\pi}{3} \), subtract \( \frac{\pi}{3} \) from \( \pi \):\[ \pi - \frac{\pi}{3} \]Express \( \pi \) and \( \frac{\pi}{3} \) to have the same denominator:\[ \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]So, the supplement of \( \frac{\pi}{3} \) is \( \frac{2\pi}{3} \).

Key Concepts

Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees or \( \frac{\pi}{2} \) radians. This concept is useful when we want to calculate an unknown angle that completes another angle to make a right angle. In mathematical terms, if you have an angle \( \theta \), its complement can be calculated by subtracting \( \theta \) from \( \frac{\pi}{2} \).
For instance, if you have an angle of \( \frac{\pi}{3} \), the complement is found by:

• Converting \( \frac{\pi}{2} \) and \( \frac{\pi}{3} \) to have a common denominator
• Subtracting the angle \( \frac{\pi}{3} \) from \( \frac{\pi}{2} \)
• The result is \( \frac{\pi}{6} \), which is the complement of \( \frac{\pi}{3} \)

This concept is simple yet powerful, allowing us to easily solve problems involving right angles.

Supplementary Angles

Supplementary angles are pairs of angles whose measures sum to 180 degrees or \( \pi \) radians. These angles form a straight line when combined. This idea is applied when determining the angle needed to create a straight angle or flat line with another angle. If you know one angle \( \theta \), the supplement is found by subtracting \( \theta \) from \( \pi \).
Here's how you find the supplement of \( \frac{\pi}{3} \):

• Match \( \pi \) and \( \frac{\pi}{3} \) to the same denominator
• Subtract \( \frac{\pi}{3} \) from \( \pi \)
• The supplement is \( \frac{2\pi}{3} \)

Understanding this principle helps in visualizing angles that add up to a complete half-circle.

Radians

Radians are an alternative to degrees for measuring angles, often used in mathematics and physics. Unlike degrees that are based on subdividing a circle into 360 pieces, radians relate directly to the properties of a circle.
A radian is defined as the angle created when the arc length is equal to the radius of the circle. Since there are \( 2\pi \) radians in a full circle (equivalent to 360 degrees), we often convert between radians and degrees.
Key conversions include:

• \( 180^{\circ} = \pi \) radians
• \( 90^{\circ} = \frac{\pi}{2} \) radians
• \( 1 \) radian \( \approx 57.3^{\circ} \)

Radians provide a direct way to work with angles in trigonometric functions, calculus, and complex numbers.

Angle Measurement

Understanding angle measurement is fundamental in geometry and trigonometry. Angles can be measured in degrees, known for everyday use, or radians, which are crucial in academic and scientific contexts.
Degrees measure angles as parts of 360, convenient for simple geometric shapes like triangles and circles. Radians, however, are more practical for advanced mathematical calculations because they relate directly to the properties of a circle. An essential part of working with angles is shifting between these two units depending on the problem.
A critical aspect of angle measurement involves:

• Recognizing that complementary angles add to \( 90^{\circ} \) or \( \frac{\pi}{2} \)
• Understanding that supplementary angles add to \( 180^{\circ} \) or \( \pi \)
• Being able to convert between radians and degrees

This knowledge aids in solving a multitude of geometric and trigonometric problems.