Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Convert these angles given in radians to degrees. (a) \(\frac{3 \pi}{4}\) (b) \(\frac{-3 \pi}{4}\) (c) \(\frac{5 \pi}{6}\) (d) \(\frac{3 \pi}{2}\) (e) \(\frac{5 \pi}{4}\) (f) \(-3.2\) (g) 4

Step-by-Step Solution

Answer

(a) 135 degrees, (b) -135 degrees, (c) 150 degrees, (d) 270 degrees, (e) 225 degrees, (f) -183.27 degrees, (g) 229.18 degrees

Step 1: Identify the Conversion Ratio

Identify the conversion ratio. The ratio is \(1 rad = \frac{180}{\pi} degrees\). This is a constant that will be multiplied with radian measures to convert them to degrees.

Step 2: Convert Each Angle

(a) For \(\frac{3 \pi}{4}\) radians, convert by multiplying by \(\frac{180}{\pi}:\) \(\frac{3 \pi}{4} * \frac{180}{\pi} = 135 degrees\). (b) For \(\frac{-3 \pi}{4}\), perform a similar conversion: \(\frac{-3 \pi}{4} * \frac{180}{\pi} = -135 degrees\). (c) \(\frac{5 \pi}{6} * \frac{180}{\pi} = 150 degrees\). (d) \(\frac{3 \pi}{2} * \frac{180}{\pi} = 270 degrees\). (e) \(\frac{5 \pi}{4} * \frac{180}{\pi} = 225 degrees\). (f) For a negative radian measure without \(\pi\), apply the conversion: \(-3.2 * \frac{180}{\pi} \approx -183.27 degrees\). (g) For positive radian measures without pi: \(4 * \frac{180}{\pi} \approx 229.18 degrees\).

Step 3: Check the Answers

Make sure the conversion was done correctly by checking if the signs are right and if the degree measures make sense. For example, an angle of \(-3.2\) radians is a little less than half a full circle in the negative direction, so \(-183.27\) degrees makes sense.

Key Concepts

Radians

Radians are a way to measure angles using the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians use the relationship between the radius and the circumference.

The entire circle is equal to \(2\pi\) radians. That means when you wrap the radius around the circle's edge, it goes around a bit over 6 times (since \(2\pi \approx 6.283\)).

Radians are particularly useful in mathematics, especially in trigonometry and calculus, because they simplify many formulas. Understanding radians allows you to grasp deeper concepts in these subjects. Remember, \(\pi\) radians is a half-circle, which is equivalent to 180 degrees.

Degrees

Degrees are a more familiar way to measure angles, especially useful in everyday life. A full circle encompasses 360 degrees.

Why 360? Historically, it may be linked to ancient calendars and the movement of stars.

When measuring angles in degrees:

• A right angle is 90 degrees.
• A straight line is 180 degrees.
• A full circle is 360 degrees.

Degrees are intuitive for many people, making them initially easier to visualize than radians. They are prevalent in fields like navigation, carpentry, and many engineering disciplines.

Conversion Ratio

Converting between radians and degrees is essential in many mathematical contexts. Fortunately, there's a simple conversion ratio: \(1 \text{ rad} = \frac{180}{\pi} \text{ degrees}\).

This ratio helps to easily switch between these units:

• To convert radians to degrees, multiply by \(\frac{180}{\pi}\).
• To convert degrees to radians, multiply by \(\frac{\pi}{180}\).

This conversion helps bridge the gap between different fields of mathematics and practical applications, ensuring clarity when interpreting angles.

Trigonometry

Trigonometry deals with the study of triangles, particularly right triangles, and the relationships between their angles and sides. It heavily involves the use of radians and degrees for angle measures.

Why Trigonometry Needs Both Units:

• In trigonometry, radians are often used for mathematical simplicity in formulas.
• Degrees are used for solving practical problems and in teaching, as they are more intuitive.

Functions like sine, cosine, and tangent rely on angle measurements, regardless of the unit. Understanding how to switch between radians and degrees enhances problem-solving skills in trigonometry, giving insights into angle behaviors and relationships.