Problem 31 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{3 \pi}{4}\)

Step-by-Step Solution

Answer

Positive coterminal: \(\frac{11\pi}{4}\), negative coterminal: \(-\frac{5\pi}{4}\).

Step 1: Understand coterminal angles

Coterminal angles are angles that share the same initial and terminal sides. To find coterminal angles, you can add or subtract multiples of the full angle circle (for radians, this is typically \(2\pi\)).

Step 2: Find a positive coterminal angle

To find a positive coterminal angle with \(\frac{3\pi}{4}\), add \(2\pi\) to \(\frac{3\pi}{4}\): \[\frac{3\pi}{4} + 2\pi = \frac{3\pi}{4} + \frac{8\pi}{4} = \frac{11\pi}{4}.\] Thus, \(\frac{11\pi}{4}\) is a positive coterminal angle.

Step 3: Find a negative coterminal angle

To find a negative coterminal angle with \(\frac{3\pi}{4}\), subtract \(2\pi\) from \(\frac{3\pi}{4}\): \[\frac{3\pi}{4} - 2\pi = \frac{3\pi}{4} - \frac{8\pi}{4} = -\frac{5\pi}{4}.\] Thus, \(-\frac{5\pi}{4}\) is a negative coterminal angle.

Key Concepts

Angle Measurement

An angle is a figure formed by two rays (called the sides of the angle) with a common endpoint known as the vertex. Angle measurement is important for both geometric and trigonometric calculations. In the world of mathematics, angles can be measured in different units.

• Degrees: One way to measure angles is in degrees. A full circle is 360 degrees.
• Radians: Another way to measure angles is in radians, where a full circle is equal to \(2\pi\) radians.

When working with angles, it's crucial to understand what unit you are using, as this can affect your calculations. For instance, an angle of 180 degrees is equivalent to \(\pi\) radians. Often, problems involving trigonometry or any kind of circular motion will express angles in radians, as it simplifies formulas and calculations.

Radian Measure

Radian measure is a unique way of measuring angles based on the radius of the circle. One radian is the angle made when the arc length is equal to the radius. It's a straight-forward and natural way to define angles in terms of the circle itself.

• A key advantage of radians is that they make mathematical formulas much cleaner and more straightforward.
• To convert from degrees to radians, multiply by \(\frac{\pi}{180}\).
• Conversely, to convert from radians to degrees, multiply by \(\frac{180}{\pi}\).

For example, the angle \(\frac{3\pi}{4}\) radians is typically encountered in the unit circle settings, making calculations involving periodic functions like sine and cosine smoother and more intuitive.

Positive and Negative Angles

Angles can be both positive and negative. The sign of the angle depends on the direction in which it is measured. This concept is very helpful for finding coterminal angles, which are angles that share the same position on the unit circle.

• Positive Angles: Measured in the counterclockwise direction from the initial side along the unit circle.
• Negative Angles: Measured in the clockwise direction. They are essentially the reverse path of their positive counterparts.

To find coterminal angles, you add or subtract multiples of \(2\pi\) (in the case of radians) to your original angle. In the exercise, adding \(2\pi\) to \(\frac{3\pi}{4}\) gave us a positive coterminal angle of \(\frac{11\pi}{4}\), while subtracting \(2\pi\) resulted in the negative coterminal angle of \(-\frac{5\pi}{4}\). Understanding this concept is a cornerstone of trigonometry and helps in determining periodicity and symmetry in trigonometric functions.