Problem 12 - Applied Calculus

Question

For each angle: i. Find the radian measure (without using a calculator). ii. Check your answers using a graphing calculator. a. \(45^{\circ}\) b. \(240^{\circ}\) c. \(-150^{\circ}\)

Step-by-Step Solution

Answer

45° is \(\frac{\pi}{4}\), 240° is \(\frac{4\pi}{3}\), -150° is \(-\frac{5\pi}{6}\).

Step 1: Understand the Conversion Formula

To convert angles from degrees to radians, we use the formula: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \). This formula helps us change any angle given in degrees to radians.

Step 2: Convert 45 Degrees to Radians

Using the formula from Step 1, we substitute 45 for degrees: \[ 45^{\circ} \times \frac{\pi}{180} = \frac{45\pi}{180} = \frac{\pi}{4} \] So, the radian measure for \( 45^{\circ} \) is \( \frac{\pi}{4} \).

Step 3: Convert 240 Degrees to Radians

Using the conversion formula for 240 degrees: \[ 240^{\circ} \times \frac{\pi}{180} = \frac{240\pi}{180} = \frac{4\pi}{3} \] Thus, the radian measure for \( 240^{\circ} \) is \( \frac{4\pi}{3} \).

Step 4: Convert -150 Degrees to Radians

Similarly, for -150 degrees: \[ -150^{\circ} \times \frac{\pi}{180} = \frac{-150\pi}{180} = -\frac{5\pi}{6} \] Hence, the radian measure for \( -150^{\circ} \) is \(-\frac{5\pi}{6} \).

Step 5: Check Calculations with a Graphing Calculator

Turn on the graphing calculator and enter the conversion of each degree measure (45, 240, -150) to radians using the calculator's conversion functionality. Verify that the outputs are \( \frac{\pi}{4} \), \( \frac{4\pi}{3} \), and \(-\frac{5\pi}{6} \) respectively.

Key Concepts

Angle Measurement

Angles can be measured in two main units: degrees and radians. Degrees are a more familiar way of measuring angles. A full circle is divided into 360 degrees, so each degree is a tiny fraction of the complete circle.
Radians provide another perspective. Instead of dividing the circle into smaller parts, the circle's radius plays a critical role.
One radian is the angle formed when the arc's length is equal to the circle's radius. Since the entire circumference is a little more than 6 times the radius (specifically, about 6.283, which is roughly \(2\pi\)), there are \(2\pi\) radians in a full circle.

• Degrees use the number 360 for a full circle.
• Radians use \(2\pi\).

Understanding both measurement units is crucial for various applications in trigonometry and calculus.
This duality is especially important when dealing with calculus where radians are often preferred due to their natural relationship with the circle.

Radian Measure

The radian measure of an angle provides a way to move seamlessly between degrees and radians using the conversion formula: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \).
This formula is essential for converting degrees, a more intuitive unit, into radians, which are more suited for advanced mathematics.
Applying the formula needs a clear understanding of basic arithmetic and fraction simplification. For example:

• \(45^{\circ}\) becomes \( \frac{\pi}{4} \) using \(45 \times \frac{\pi}{180} \)
• \(240^{\circ}\) follows as \( \frac{4\pi}{3} \) from \(240 \times \frac{\pi}{180} \)
• \(-150^{\circ}\) results in \(-\frac{5\pi}{6} \) using \(-150 \times \frac{\pi}{180} \)

Once you memorize this conversion equation, switching between units becomes straightforward, aiding in better understanding and more practical calculations in complex mathematical problems.

Graphing Calculator

A graphing calculator is a powerful tool that can rapidly verify manual calculations, including angle conversions. Most modern graphing calculators have built-in functions to change degrees to radians and vice versa.
To use this feature, enter the degree measure, and through the calculator's menu, select the conversion option.
For instance, to confirm our manual calculations:

• Input \(45^{\circ}\) and check it outputs \(\frac{\pi}{4}\).
• Enter \(240^{\circ}\) ensuring it shows \(\frac{4\pi}{3}\).
• Verify \(-150^{\circ}\) converts to \(-\frac{5\pi}{6}\).

Using a graphing calculator not only saves time but also reinforces the learning process, providing an immediate check for your work.
These devices are especially useful in exams and complex calculations where quick, accurate results are needed. Mastering their functions can significantly enhance mathematical proficiency.