Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Convert each degree measure to radians. Round to the nearest hundredth. $$39^{\circ}$$

Step-by-Step Solution

Answer

39° is approximately 0.68 radians.

Step 1: Understand the Conversion Formula

To convert degrees to radians, we use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This formula is based on the fact that \( 180^{\circ} \) is equal to \( \pi \) radians.

Step 2: Substitute the Degree Value

Substitute \( 39^{\circ} \) into the formula: \( 39 \times \frac{\pi}{180} \).

Step 3: Calculate the Fraction

Calculate \( \frac{39}{180} \). This simplifies to approximately \( 0.2167 \).

Step 4: Multiply by Pi

Now multiply \( 0.2167 \) by \( \pi \) to convert to radians: \( 0.2167 \times \pi \).

Step 5: Round the Result

Calculate \( 0.2167 \times \pi \) to get approximately \( 0.6807 \). Round this to the nearest hundredth to get \( 0.68 \).

Key Concepts

Radian Measure

Radian measure is a way to express angles using the radius of a circle. Instead of degrees, which are a common unit, radians provide an alternative that relates directly to the properties of a circle. The entire circumference of a circle is divided by its radius to describe the rotation more naturally.For a full circle, which is 360 degrees, the radian measure is exactly \( 2\pi \) radians. Thus, a half circle is \( \pi \) radians, and a quarter-circle is \( \frac{\pi}{2} \) radians.

Here's why radians are useful:

• They allow for a more straightforward mathematical formulation of derivatives and integrals in calculus.
• Radian measure correlates directly with arc lengths, which are crucial in trigonometry and geometry.

Understanding radians helps in converting degree measurements in order to solve problems involving rotations and circle-based calculations.

Angle Conversion

Converting angles from degrees to radians is an essential skill in mathematics, especially in trigonometry.To perform the conversion, remember the key relationship: \( 180^{\circ} = \pi \text{ radians} \).This means that 1 degree is equal to \( \frac{\pi}{180} \text{ radians} \). Therefore, to convert any angle given in degrees to radians, you multiply the degree measure by \( \frac{\pi}{180} \).

For example, to convert 39 degrees to radians:

• Calculate \( 39 \times \frac{\pi}{180} \).
• This simplifies to \( \frac{39\pi}{180} \).
• By dividing, you arrive at approximately \( 0.2167\pi \).
• Multiply by \( \pi \) to find the radian measure.
• Finally, round your result to a practical decimal, here approximately \( 0.68 \) radians.

Following these steps makes angle conversions intuitive.

Trigonometry

Trigonometry is the branch of mathematics that deals with angles and their relationships to functions like sine, cosine, and tangent.It's fundamental in understanding periodic phenomena, such as sound waves, light waves, and alternating currents.
In trigonometry, the unit circle is often used, where the radius is 1, and the angle's measure in radians directly expresses the length of its arc. Radians provide a direct correlation between an angle and the arc length it subtends on the unit circle, allowing trigonometric functions to take on a more natural and simplified form.

Here's why understanding trigonometry in radians is helpful:

• Trigonometric functions in calculus generally prefer radian measure for derivatives and integrals, as they simplify to forms like \( \sin(x) \rightarrow \cos(x) \).
• Radians simplify learning about the properties of waves and oscillations because they provide a smooth and regular framework.
• Understanding radians and their conversion sooner helps grasp circular motion and harmonic motion problems more effectively.

Grasping trigonometry with a solid understanding of radians will enhance problem-solving skills in physics and engineering.