Problem 10 - Applied Calculus

Question

For each angle: i. Find the radian measure (without using a calculator). ii. Check your answers using a graphing calculator. a. \(15^{\circ}\) b. \(210^{\circ}\) c. \(-90^{\circ}\)

Step-by-Step Solution

Answer

a. \( \frac{\pi}{12} \); b. \( \frac{7\pi}{6} \); c. \(-\frac{\pi}{2}\).

Step 1: Understanding Degree to Radian Conversion

To convert an angle from degrees to radians, use the conversion factor \( \frac{\pi}{180^{\circ}} \). This factor arises because \( 180^{\circ} \) is equal to \( \pi \) radians.

Step 2: Convert 15 Degrees to Radians

Multiply 15 by the conversion factor: \[ 15^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{15\pi}{180} = \frac{\pi}{12} \] Thus, the radian measure of \( 15^{\circ} \) is \( \frac{\pi}{12} \).

Step 3: Convert 210 Degrees to Radians

Multiply 210 by the conversion factor: \[ 210^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{210\pi}{180} = \frac{7\pi}{6} \] Thus, the radian measure of \( 210^{\circ} \) is \( \frac{7\pi}{6} \).

Step 4: Convert -90 Degrees to Radians

Multiply -90 by the conversion factor: \[ -90^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{-90\pi}{180} = -\frac{\pi}{2} \] Thus, the radian measure of \(-90^{\circ} \) is \(-\frac{\pi}{2} \).

Step 5: Verify with a Calculator

Use a graphing calculator to confirm each conversion:1. \( 15^{\circ} \) should give \( \frac{\pi}{12} \approx 0.2618 \) radians.2. \( 210^{\circ} \) should give \( \frac{7\pi}{6} \approx 3.6652 \) radians.3. \( -90^{\circ} \) should give \( -\frac{\pi}{2} \approx -1.5708 \) radians.

Key Concepts

Angle Measurement

Angle measurement can be understood as the amount of rotation needed to bring one line or plane into coincidence with another. In geometry and trigonometry, we use two main units to measure angles: degrees and radians. Degrees are more familiar and are used in everyday contexts, like measuring the angles of a triangle. A full circle is divided into 360 degrees, with each degree representing 1/360th of a full rotation.

Radians, however, are more natural for mathematical calculations, especially in trigonometry. A radian measures the length of an arc created by an angle, where the arc length is equal to the radius of the circle. This makes radians immensely useful in calculus and other advanced areas of mathematics. Understanding how to convert between these two units is crucial for tackling various mathematical problems.

Radian Measure

Radian measure is a way of quantifying angles based on the arc length created by that angle on a unit circle (a circle with a radius of one unit). One full revolution around a circle corresponds to an angle of \(2\pi\) radians, which is equivalent to \(360^{\circ}\). This means \(\pi\) radians corresponds to \(180^{\circ}\).

To convert degrees to radians, we use the conversion factor \(\frac{\pi}{180}\). Here's why:

• \(180^{\circ} = \pi\) radians
• Therefore, 1 degree is \(\frac{\pi}{180}\) radians

To apply this conversion, multiply the degree measure by \(\frac{\pi}{180}\). For instance, converting \(15^{\circ}\) to radians involves calculating \(15 \times \frac{\pi}{180} = \frac{\pi}{12}\). Similarly, \(210^{\circ}\) becomes \(\frac{7\pi}{6}\), and \(-90^{\circ}\) converts to \(-\frac{\pi}{2}\). Recognizing these common conversions will enhance your comfort with angle measurements in different formats.

Trigonometry Basics

Trigonometry is the study of relationships between angles and sides of triangles, often relying heavily on angle measurements in both degrees and radians. Understanding both units is essential because many trigonometric functions and formulas use radians.

The primary trigonometric functions—sine, cosine, and tangent—all relate to angles and are widely used in various applications, from physics to engineering. Here are some key takes:

• Functions take input as an angle (in degrees or radians) and provide output that relates to the triangle's sides.
• Using radians simplifies derivatives and integrals, common in calculus-related applications.
• Knowing how to switch between degrees and radians helps solve problems naturally in the format required.

These fundamentals establish a firm foundation for broader math and science topics, emphasizing the significance of understanding radian and angle conversions.