Problem 80 - Precalculus

Question

Determine the angle of the smallest possible positive measure that is coterminal with each of the angles whose measure is given. Use degree or radian measures accordingly. $$42$$

Step-by-Step Solution

Answer

The smallest positive coterminal angle is 42 degrees.

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find coterminal angles, we either add or subtract full rotations. A full rotation is 360 degrees.

Step 2: Identify the Given Angle

The given angle is 42 degrees. This angle is already positive and less than 360 degrees.

Step 3: Confirm the Smallest Positive Coterminal Angle

Since 42 degrees is already positive and falls within the range of 0 to 360 degrees, it is indeed the smallest positive coterminal angle with itself.

Key Concepts

Coterminal Angles

In trigonometry, understanding coterminal angles is important. Coterminal angles are those that share the same initial and terminal sides, meaning they end up in the same position on a circle, even if they have different angle measures.

There are infinite coterminal angles for any given angle, and they are found by adding or subtracting full rotations. A full rotation in a circle is 360 degrees. Therefore, to find additional coterminal angles:

• Add or subtract 360 degrees (or multiples of 360) to the original angle.
• For radians, add or subtract \(2\pi\) (or multiples of \(2\pi\)).

This concept helps in simplifying trigonometric functions, transitioning between angles, and solving problems.

For instance, the angle of 42 degrees is already within the range of 0 to 360 degrees, making it its smallest positive coterminal angle.

Angle Measurement

Angle measurement can be expressed in several ways, but two of the most common units are degrees and radians.

A degree is a unit of measurement that represents 1/360th of a full rotation around a circle. Therefore, a full circle is 360 degrees. This method is often intuitive and widely used in everyday contexts.

Radians, on the other hand, are based on the concept of the radius of a circle. One radian is the angle formed when the length of the arc is equal to the radius of the circle. Therefore, a full circle is \(2\pi\) radians, which is approximately 6.28318 radians.

• Radians are often used in higher mathematics due to their natural relationship to the properties of circles and the unit circle.
• Degrees provide a simpler numerical answer for many practical applications.

Choosing the appropriate unit depends on the context of the problem. Understanding both is crucial to mastering trigonometry.

Degrees vs. Radians

Degrees and radians are both units for measuring angles, each with its uses and advantages. The conversion between these two units is key for solving various trigonometric problems.

To convert degrees to radians, use the conversion factor \(\frac{\pi}{180}\). For example, to convert 45 degrees to radians, multiply:

• \(45 \times \frac{\pi}{180} = \frac{\pi}{4}\) radians

Similarly, converting radians to degrees involves multiplying by \(\frac{180}{\pi}\):

• \(\pi\) radians converts to \(\pi \times \frac{180}{\pi} = 180\) degrees

• Degrees are often used in navigation, surveying, and many practical everyday applications.
• Radians simplify calculations in calculus and other more advanced mathematics because they relate directly to the properties of the circle.

Mastering the conversion ensures flexibility in working with different problems, enhancing understanding and problem-solving abilities.