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Chapter 4: Problem 61

Convert each of the following to radians without using a calculator. \(150^{\circ}\)

Short Answer

Expert verified
The angle \(150^{ ext{degrees}}\) converts to \(\frac{5\pi}{6}\) radians.

Step by step solution

01

Understand the Problem

We need to convert the angle measurement from degrees to radians. The general formula for converting degrees to radians is \( ext{radians} = rac{ ext{degrees} imes ext{pi}}{180}\).
02

Substitute the Given Value

Substitute the given angle of \(150^{ ext{degrees}}\) into the formula: \( ext{radians} = rac{150 imes ext{pi}}{180} \).
03

Simplify the Fraction

Now simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 30: \( rac{150 imes ext{pi}}{180} = rac{5 imes ext{pi}}{6} \).
04

Write the Result in Radians

The converted angle in radians is \( rac{5 ext{pi}}{6} \).

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Key Concepts

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

Radians
Radians are a unit of angle measurement used in mathematics, particularly in trigonometry. They provide a different way of expressing angles compared to degrees. Instead of splitting a circle into 360 equal parts, radians use the circle's radius to measure how far around the circle a point travels.
  • One radian is the angle created when the radius is wrapped along the edge of the circle.
  • This means that the entire circumference of a circle is equal to 2蟺 radians.
  • Since a circle is 360 degrees, 2蟺 radians is equivalent to 360 degrees.
By understanding radians, one can easily transition between linear and angular measurements, which is particularly useful in calculus and physics.
Degrees
Degrees are perhaps the most familiar method for measuring angles. The degree system divides a complete circle into 360 equal parts, where each part is one degree. This system is ancient and widely used in various applications, from navigation to geometry.
  • There are 360 degrees in a circle.
  • This makes it easy to express parts of a circle; for example, half a circle is 180 degrees.
  • Degrees are intuitive and often used in everyday experiences.
Knowing how to measure and convert degrees is essential because many practical problems involve this unit. While radians are sometimes more convenient mathematically, degrees remain very popular due to their simplicity.
Conversion Formula
The conversion formula for changing degrees to radians is a crucial tool in mathematics, especially for those dealing with trigonometric functions. The formula is given by:\[\text{radians} = \frac{\text{degrees}\times \pi}{180}\]This equation establishes the relationship between the two units and allows easy conversion.
  • The number 180 arises because a semicircle measures 180 degrees, which corresponds to 蟺 radians.
  • This relationship means that 蟺 radians equal 180 degrees.
  • To convert from degrees to radians, you multiply the number of degrees by 蟺 and then divide by 180.
Using this formula, you can convert any angle in degrees to radians easily, as seen in the example of converting 150 degrees to \(\frac{5\pi}{6}\).
Trigonometry
Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. It plays a critical role in understanding angles measured in both degrees and radians.
  • Trigonometric functions such as sine, cosine, and tangent often rely on angle measures.
  • Whether working in degrees or radians, these functions remain consistent, but the angle measure must be properly converted.
  • In calculus, radians are typically more convenient because they simplify derivative calculations of trigonometric functions.
Understanding how to convert between radians and degrees is essential for solving trigonometric problems accurately, ensuring that these functions work correctly in a wide range of mathematical and physical applications.

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Most popular questions from this chapter

Evaluate each of the following if \(x\) is \(\pi / 2\) and \(y\) is \(\pi / 6\). \(\cos x+\cos y\)

Evaluate without using a calculator. $$ \tan ^{-1}\left(\tan \frac{5 \pi}{6}\right) $$

Graph one complete cycle for each of the following. In each case, label the axes so that the amplitude and period are easy to read. \(y=-2 \sin (-3 x), 0 \leq x \leq 2 \pi\)

The following problems review material we covered in Section 3.1. Name the reference angle for each angle below.\(321^{\circ}\)

Graph one complete cycle of each of the following. In each case, label the axes accurately and identify the amplitude for each graph. \(y=-4 \sin x\)
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