Problem 11 - Applied Calculus

Question

For each angle: i. Find the radian measure (without using a calculator). ii. Check your answers using a graphing calculator. a. \(60^{\circ}\) b. \(315^{\circ}\) c. \(-120^{\circ}\)

Step-by-Step Solution

Answer

a. \(\frac{\pi}{3}\); b. \(\frac{7\pi}{4}\); c. \(-\frac{2\pi}{3}\).

Step 1: Understanding Degree to Radian Conversion

To convert degrees into radians, use the formula \( Radians = Degrees \times \left(\frac{\pi}{180^{\circ}}\right) \).This is because \(180^{\circ}\) is equivalent to \(\pi\) radians.

Step 2: Convert 60 Degrees to Radians

Using the formula: \[ 60^{\circ} \times \left(\frac{\pi}{180^{\circ}}\right) = \frac{\pi}{3} \].Therefore, the radian measure for \(60^{\circ}\) is \(\frac{\pi}{3}\).

Step 3: Convert 315 Degrees to Radians

Apply the conversion formula: \[ 315^{\circ} \times \left(\frac{\pi}{180^{\circ}}\right) = \frac{7\pi}{4} \].Hence, \(315^{\circ}\) is equivalent to \(\frac{7\pi}{4}\) radians.

Step 4: Convert -120 Degrees to Radians

Use the formula to convert:\[ -120^{\circ} \times \left(\frac{\pi}{180^{\circ}}\right) = -\frac{2\pi}{3} \].Thus, the radian measure for \(-120^{\circ}\) is \(-\frac{2\pi}{3}\).

Step 5: Verification with Graphing Calculator

Input each degree measure into a graphing calculator and convert to radians to verify that \(60^{\circ}\) is \(\frac{\pi}{3}\), \(315^{\circ}\) is \(\frac{7\pi}{4}\), and \(-120^{\circ}\) is \(-\frac{2\pi}{3}\). The calculator should confirm these conversions are correct.

Key Concepts

Radian Measure

Radian measure is a way to express angles. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of a circle. One radian equals the angle created when the arc length is the same as the radius of the circle. It's a natural way to measure angles because it directly relates the angle to the size of the circle.
In mathematical terms, the full circle is expressed as \(2\pi\) radians. That means half a circle, or 180 degrees, is \(\pi\) radians. This relationship of degrees to radians is crucial in mathematics and applications, particularly in calculus and physics.
To convert degrees into radians, we use the conversion factor of \(\frac{\pi}{180^{\circ}}\). This tells us how many radians one degree is and helps in converting any degree measure into radians easily.

Degree Measure

Degrees are a more familiar way of measuring angles in everyday life. There are 360 degrees in a circle, which makes degrees a very manageable and commonly understood unit. Each degree represents \(\frac{1}{360}\) of a complete circle.
When working with trigonometry, often you'll need to switch between degrees and radians. This is because some trigonometric functions and calculus operations are easier to compute or understand in radians. However, like learning any new concept, converting degrees into a different form, such as radians, initially can be a bit challenging.
Degrees are perfect for simple, day-to-day measurements, but understanding their relationship to radians is vital for advanced mathematics.

Trigonometric Conversions

Trigonometric conversions allow us to translate between different systems of angle measurement \(\) - degrees and radians.

• When you see an angle measured in degrees and wish to work with trigonometric functions, conversion to radians is often the first step.
• The formula \(Radians = Degrees \times \(\frac{\pi}{180^{\circ}}\)\) simplifies this conversion process.
• This is crucial for functions, like sine or cosine, that require input in radians for certain calculations or when integrating trigonometric functions in calculus.

After converting into radians, verifying your calculations with a graphing calculator can be reassuring. This provides double-check assurance that your conversion is accurate, particularly when working with complex angle values or negative degrees, as in the case with angles like \(-120^{\circ}\).
Mastering these conversions will enhance your understanding and efficiency in trigonometry and calculus, making you ready for more advanced mathematical problems.