Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine two coterminal angles in radian measure (one positive and one negative) for each angle. (There are many correct answers). (a) \(\frac{9 \pi}{4}\) (b) \(-\frac{2 \pi}{15}\)

Step-by-Step Solution

Answer

The positive and negative coterminal angles for the given angles \(\frac{9 \pi}{4}\) and \(-\frac{2 \pi}{15}\) are: For \(\frac{9 \pi}{4}\), Positive coterminal angle = \(\theta_1 = \frac{\pi}{4}\) Negative coterminal angle = \(\theta_2 = \frac{9\pi}{4} + 2\pi\) For \(-\frac{2 \pi}{15}\), Positive coterminal angle = \(\theta_3 = 2\pi - \frac{2\pi}{15}\) Negative coterminal angle = \(\theta_4 = -\frac{2\pi}{15} - 2\pi\)

Step 1 - Compute coterminal angles for \(\frac{9 \pi}{4}\)

Subtract \(2\pi\) from \(\frac{9\pi}{4}\) and check if it equals a number less than \(2\pi\), in other words, subtract until we get an angle smaller than 2\(\pi\). Let's call this angle \(\theta_1\). For the negative coterminal angle, add \(2\pi\) to \(\frac{9\pi}{4}\). Let's call this angle \(\theta_2\).

Step 2 - Compute coterminal angles for \(-\frac{2 \pi}{15}\)

Add \(2\pi\) to \(-\frac{2\pi}{15}\) and check if it equals a number less than \(2\pi\) but greater than 0, in other words, add until we get an angle between 0 and \(2\pi\). Let's call this angle \(\theta_3\). For the negative coterminal angle, subtract \(2\pi\) from \(-\frac{2\pi}{15}\). Let's call this angle \(\theta_4\).

Key Concepts

Radian Measure

Radian measure is a way of expressing angles. Unlike degrees, which divide a circle into 360 parts, radians are a bit more abstract. A full circle is always equal to \(2\pi\) radians. Think of a radian as the angle made when you wrap the radius of a circle along its circumference. This is helpful since we often deal with circles in trigonometry and calculus.

To convert an angle from degrees to radians, you can use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

Because radians tie directly to the properties of circles, they often make calculations involving circular motion smoother. You'll often encounter radians when looking for coterminal angles, which are angles that share the same terminal side in standard position.

Positive Angle

A positive angle typically means you are measuring counterclockwise from the positive x-axis. This is a universal convention in mathematics. As you determine positive coterminal angles, you take your given angle and add one or more full circle rotations (\(2\pi\)) until the result is greater than zero and often less than \(2\pi\).

• For example, with \(\frac{9\pi}{4}\), by subtracting \(2\pi\), you find a positive coterminal angle.
• Positive angles are common in real-world scenarios, such as calculating rotations or determining angular paths.

Understanding positive angles leads to better grasp on rotational motion and helps to visualize mathematical problems involving cycles.

Negative Angle

Negative angles are essentially the opposite of positive angles, often measured clockwise from the positive x-axis. When you search for a negative coterminal angle, you subtract \(2\pi\) (a full rotation) from the given angle multiple times if necessary.

• For instance, for \(-\frac{2\pi}{15}\), adding \(2\pi\) gives you a positive coterminal angle, but you can subtract \(2\pi\) again if needed to find a negative one.
• Negative angles are particularly useful when dealing with reversing paths or describing motions that go back and forth.

Understanding negative angles is key in physics and engineering, where opposite directions are frequently accounted for in design and calculations.