Problem 52 - Precalculus with Calculus Previews

Question

In Problems \(47-52,\) find the angle between 0 and \(2 \pi\) that is coterminal with the given angle. $$ 7.5 $$

Step-by-Step Solution

Answer

The coterminal angle between 0 and \(2\pi\) is approximately \(1.2168\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides but differ by full rotations. To find a coterminal angle within a specific range, adjust the given angle by adding or subtracting full circles (multiples of \(2\pi\)).

Step 2: Determine Full Rotation

Since we are working in radians, one full rotation is \(2\pi\). We will determine how modifying the angle \(7.5\) affects its position relative to 0 and \(2\pi\). Indeed, \(7.5\) is outside this range.

Step 3: Subtract Whole Circles

Subtract \(2\pi\) from \(7.5\) to bring it within the desired range. So, \(7.5 - 2\pi = 7.5 - 6.2832 \approx 1.2168\).

Step 4: Verification

Check if \(1.2168\) lies between 0 and \(2\pi\). Since \(1.2168 > 0\) and \(1.2168 < 2\pi\), it is the correct coterminal angle within the given range.

Key Concepts

Radians: The Angle Unit

Radians are a unit of angular measure used extensively in mathematics and engineering. Unlike degrees, which divide a circle into 360 parts, radians use the circle's radius to define its angle. A full circle is equivalent to an angle of \(2\pi\) radians. This unique measurement system allows for easier integration with various mathematical formulas, especially when dealing with trigonometric functions.

• Degree to radian conversion involves multiplying by \(\frac{\pi}{180}\)
• Radian to degree conversion uses \(\frac{180}{\pi}\)

Keep in mind, an angle's value in radians provides a direct connection to arc length and radius, enabling more efficient calculations in various fields.

Full Rotation: Completing the Circle

A full rotation refers to a complete turn around a circle. In radians, this full circular rotation is quantified as \(2\pi\). This is key when working with coterminal angles, since you can add or subtract full rotations to or from any angle to keep it equivalent.Coterminal angles are those which differ by one or more full rotations. To find a coterminal angle, simply:

• Add or subtract \(2\pi\) from your angle
• Repeat the process to find multiple coterminal angles if needed

Whether dealing with angle measures or coordinates, understanding full rotations is crucial to many mathematical applications.

Angle Conversion: Finding Coterminal Angles

Angle conversion is about transforming an angle to a different expression or finding its equivalent in a specified range. Coterminal angles are converted by adjusting the angle value by the full rotations required to fit into the desired range. In the exercise example above, we converted 7.5 by subtracting \(2\pi\) (which equals 6.2832) to place the angle between 0 and \(2\pi\).Steps to convert an angle to its coterminal form:

• Identify the range for the result (e.g., 0 to \(2\pi\) in radians)
• Add or subtract \(2\pi\) until the angle fits within this range

Converting angles this way allows you to directly apply them in other calculations or geometric interpretations.

Trigonometric Concepts: Understanding the Base

Trigonometry involves studying angles, their measures, and their relationships with circles. Fundamental trigonometric functions such as sine, cosine, and tangent rely on angle measures, often expressed in radians. Trigonometric concepts are essential in understanding how angles function in two-dimensional space, and how they repeat at intervals of \(2\pi\) due to periodic nature.Key trigonometric functions include:

• Sine (\(\sin\)): relates to the y-coordinate of the point on the unit circle
• Cosine (\(\cos\)): relates to the x-coordinate of the point on the unit circle
• Tangent (\(\tan\)): the ratio of \(\sin\) to \(\cos\)

Understanding these functions and their cyclic behavior is crucial for solving real-world problems that involve periodic events, such as sound waves or pendulum motion.