Problem 50 - Precalculus with Calculus Previews

Question

In Problems \(47-52,\) find the angle between 0 and \(2 \pi\) that is coterminal with the given angle. $$ -\frac{9 \pi}{5} $$

Step-by-Step Solution

Answer

The coterminal angle is \(\frac{\pi}{5}\).

Step 1: Understand Coterminal Angles

Coterminal angles can be found by adding or subtracting multiples of \(2\pi\) to the given angle. These angles share the same initial and terminal sides.

Step 2: Determine the Initial Angle

The given angle is \(-\frac{9\pi}{5}\). We need to find an equivalent angle between 0 and \(2\pi\).

Step 3: Find the Suitable Multiple of \(2\pi\)

Since the angle is negative, we'll add \(2\pi\) (or multiples of \(2\pi\)) to bring it within the range of 0 to \(2\pi\). The next logical step is to calculate whether adding \(2\pi\) makes the angle non-negative.

Step 4: Calculate \((-\frac{9\pi}{5}) + 2\pi\)

Add \(2\pi\) to the angle: \(-\frac{9\pi}{5} + 2\pi\). Convert \(2\pi\) into a fraction with a denominator of 5 to combine: \(2\pi = \frac{10\pi}{5}\).

Step 5: Simplify the Expression

Simplify the expression: \(-\frac{9\pi}{5} + \frac{10\pi}{5} = \frac{\pi}{5}\). This angle is between 0 and \(2\pi\).

Step 6: Verify the Final Answer

Ensure that the resulting angle, \(\frac{\pi}{5}\), is indeed between 0 and \(2\pi\). Since \(0 < \frac{\pi}{5} < 2\pi\), \(\frac{\pi}{5}\) is valid.

Key Concepts

Angle Measurements

Angles are pivotal in geometry, trigonometry, and even in our daily lives. They measure the space between two intersecting lines or surfaces. In mathematics, angles are measured in degrees or radians. Radians are the standard unit for measuring angles in trigonometry. This unit relates the angle to the radius of a circle. For example, a full circle is 360 degrees or \(2\pi\) radians.Understanding radians is crucial for solving problems involving circles and periodic functions. Some common radian measures are:

• \(\pi/2\) radians, which is 90 degrees
• \(\pi\) radians, which is 180 degrees
• \(2\pi\) radians, which represents a full circle or 360 degrees

Using radians helps in finding coterminal angles, as they offer a mathematical convenience that degrees sometimes lack.

Trigonometric Functions

Trigonometric functions are essential for studying angles and their relationships in mathematics. These functions include sine, cosine, and tangent, among others. They allow us to relate angles to side lengths in right triangles or describe periodic phenomena like waves.These functions have properties that are useful in solving geometric problems:

• Sine (sin): Relates to the opposite side over the hypotenuse in a right triangle.
• Cosine (cos): Concerns the adjacent side over the hypotenuse.
• Tangent (tan): Involves the opposite side over the adjacent side.

Each function cycles through values as angles increase. This cyclical nature is why \(2\pi\) radians, or a full rotation, results in the same values for sine or cosine. Thus, knowledge of these patterns aids in identifying coterminal angles, as these angles share the same sine, cosine, and tangent values.

Angle Conversion

Angle conversion is the process of changing angle measurement units from degrees to radians or vice versa. This is fundamental in trigonometry, where radians are often preferred.To convert degrees to radians:

• Multiply the degree measure by \(\pi/180\).
• For instance, 90 degrees is \(90 \times (\pi/180) = \pi/2\) radians.

To convert radians to degrees:

• Multiply the radian measure by \(180/\pi\).
• For example, \(\pi/3\) radians equals \((\pi/3) \times (180/\pi) = 60\) degrees.

This conversion is pivotal in exercises requiring angle measurements in radians. Understanding both systems and converting between them supports solving problems with coterminal angles, as these often require a trigonometrically convenient unit like radians.