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Chapter 8: Problem 2

Convert the following to radian measure. $$18^{\circ}, 72^{\circ}, 150^{\circ}$$

Short Answer

Expert verified
18掳 = \( \frac{\pi}{10} \) radians, 72掳 = \( \frac{2\pi}{5} \) radians, 150掳 = \( \frac{5\pi}{6} \) radians

Step by step solution

01

Understanding Degrees to Radians Conversion

To convert degrees to radians, use the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This means multiplying the degree measure by \( \frac{\pi}{180} \).
02

Convert 18掳 to Radians

Using the formula, convert 18掳: \( 18^{\circ} \times \frac{\pi}{180} = \frac{18\pi}{180} = \frac{\pi}{10} \). Thus, 18掳 equals \( \frac{\pi}{10} \) radians.
03

Convert 72掳 to Radians

Using the formula, convert 72掳: \( 72^{\circ} \times \frac{\pi}{180} = \frac{72\pi}{180} = \frac{2\pi}{5} \). Thus, 72掳 equals \( \frac{2\pi}{5} \) radians.
04

Convert 150掳 to Radians

Using the formula, convert 150掳: \( 150^{\circ} \times \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6} \). Thus, 150掳 equals \( \frac{5\pi}{6} \) radians.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degrees
Degrees are a way to measure angles. When you think of a full circle, it contains 360 degrees. This is because the number 360 has many factors, making it easier for computations.

To break it down:
  • An angle of 0 degrees means there is no rotation.
  • A right angle is 90 degrees.
  • A complete turn around a circle is 360 degrees.
The degree symbol is noted by a small circle (掳). So, an angle of 75 degrees is written as 75掳.

Knowing how to convert degrees is essential when you want to compare angles or work with other measurement systems.
Radians
Radians offer another way to measure angles. If you imagine wrapping the radius of a circle around its edge, the length for one wraparound is called a radian.

In a full circle, there are about 6.283 radians (which is another way to say 2蟺 radians). So:
  • A full circle is 2蟺 radians.
  • A straight angle (half-circle) is 蟺 radians.
  • A right angle (quarter-circle) is 蟺/2 radians.
Radians are particularly useful in mathematics and physics because they simplify many formulas.

Rather than dealing with degrees, using radians can often make calculations cleaner and more intuitive.
Pi (蟺)
Pi (蟺) is a very special number in mathematics. It represents the ratio of the circumference of a circle to its diameter. No matter the size of the circle, this ratio is always 蟺, around 3.14159.

Pi is an irrational number, meaning it never ends and never repeats. Here are some fun facts:
  • 蟺 is used in many formulas, especially those involving circles and waves.
  • It's named after the Greek letter 蟺.
  • Archimedes was one of the first to calculate 蟺 accurately.
Understanding 蟺 is important when working with angles and trigonometry.

Remember, converting degrees to radians uses 蟺 directly in the formula, making it an essential part of these calculations.

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Most popular questions from this chapter

Evaluate the following integrals. $$\int_{-\pi / 8}^{\pi / 8} \sec ^{2}\left(x+\frac{\pi}{8}\right) d x$$

Differentiate (with respect to \(t\) or \(x\) ): $$y=(1+\tan 2 t)^{3}$$

Find \(t\) such that \(-\pi / 2 \leq t \leq \pi / 2\) and \(t\) satisfies the stated condition. $$\sin t=\sin (7 \pi / 6)$$

Determine the value of \(\cos t\) when \(t=5 \pi,-2 \pi, 17 \pi / 2\) \(-13 \pi / 2.\)

Find \(t\) such that \(-\pi / 2 \leq t \leq \pi / 2\) and \(t\) satisfies the stated condition. $$\sin t=\sin (-4 \pi / 3)$$
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