Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Convert these angles to radian measure. (a) \(-60^{\circ}\) (b) \(45^{\circ}\) (c) \(-270^{\circ}\) (d) \(40^{\circ}\) (e) \(-120^{\circ}\)

Step-by-Step Solution

Answer

The radian equivalents of the given angles are: (a) -60° = \(-\frac{\pi }{3}\) radians, (b) 45° = \(\frac{\pi }{4}\) radians, (c) -270° = \(-\frac{3\pi }{2}\) radians, (d) 40° = \(\frac{2\pi }{9}\) radians, (e) -120° = \(-\frac{2\pi }{3}\) radians.

Step 1: Define the Conversion Factor

The conversion factor between degrees and radians is \( \frac{\pi }{180} \). This means that \( \pi \) radians are equal to 180°. Therefore, to convert degrees to radians, every degree value will be multiplied by this conversion factor.

Step 2: Convert -60 degrees to radians

To convert -60° to radians, multiply -60 by \( \frac{\pi }{180} \). The result is \(-60 \times \frac{\pi }{180} = -\frac{\pi }{3} \) radians.

Step 3: Convert 45 degrees to radians

To convert 45° to radians, multiply 45 by \( \frac{\pi }{180} \). The result is \(45 \times \frac{\pi }{180} = \frac{\pi }{4} \) radians.

Step 4: Convert -270 degrees to radians

To convert -270° to radians, multiply -270 by \( \frac{\pi }{180} \). The result is \(-270 \times \frac{\pi }{180} = -\frac{3\pi }{2} \) radians.

Step 5: Convert 40 degrees to radians

To convert 40° to radians, multiply 40 by \( \frac{\pi }{180} \). The result is \(40 \times \frac{\pi }{180} = \frac{2\pi }{9} \) radians.

Step 6: Convert -120 degrees to radians

To convert -120° to radians, multiply -120 by \( \frac{\pi }{180} \). The result is \(-120 \times \frac{\pi }{180} = -\frac{2\pi }{3} \) radians.

Key Concepts

Radian Measure

Radian measure is a different way of expressing angles that is based on the radius of a circle. Unlike degrees, which are based on dividing a circle into 360 equal parts, a radian is determined by the length of the arc that the angle subtends. Specifically, one radian is the angle created when the arc length is equal to the radius of the circle.

To convert from degrees to radians, we leverage the relationship that 180 degrees is equal to \(\pi\) radians. So, the conversion factor is \(\frac{\pi}{180}\). For example, converting -60 degrees involves multiplying -60 by the conversion factor, resulting in \( -60 \times \frac{\pi}{180} = -\frac{\pi}{3} \) radians. This calculation applies the understanding that angles can be positive or negative based on their direction of rotation, with positive angles measured counterclockwise from the horizontal axis and negative angles measured clockwise.

Unit Circle

The unit circle is a fundamental concept in trigonometry and is a circle with a radius of one, usually centered at the origin of a coordinate plane. It serves as a tool to visualize angle measures and relate them to the values of trigonometric functions.

Every angle on the unit circle corresponds to a certain point on the circle's circumference, and these points are often expressed in coordinates that correspond to the values of the cosine and sine for that angle. For instance, an angle of 45 degrees (or \(\frac{\pi}{4}\) radians) on the unit circle corresponds to the point \( (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}) \), which also represents the cosine and sine of that angle.

Understanding the unit circle is crucial for solving problems in trigonometry, as it provides a visual and conceptual way to comprehend how angles and the trigonometric functions are interrelated.

Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles, particularly right triangles. The most common functions used in trigonometry are sine, cosine, and tangent, which are used to relate an angle with ratios of two sides of a right triangle.

These relationships are constant and can be applied to any size of the right triangle, which is why they're so powerful in calculations across different fields such as physics, engineering, and geography. With the unit circle in mind, these functions are not only confined to right triangles but extend to define the values for all angles, negative or positive, and even beyond 360 degrees or \(2\pi\) radians.

Mastering trigonometry often begins with understanding the unit circle and how to convert degrees to radians, as these form the basis for further study of trigonometric functions and their applications in various complex scenarios and problems.