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Chapter 7: Problem 5

Find the exact radian measure of the angle given in degree measure. $$135^{\circ}$$

Short Answer

Expert verified
The angle is \(\frac{3\pi}{4}\) radians.

Step by step solution

01

Identify the Conversion Formula

To convert an angle from degrees to radians, we use the conversion formula: \(\theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \) . This formula relates degrees to radians based on the fact that \(180^{\circ} = \pi\) radians.
02

Apply the Formula to 135掳

Using the conversion formula, substitute \(135^{\circ}\) for \(\theta_{degrees}\): \(135^{\circ} \times \frac{\pi}{180}\).
03

Simplify the Expression

First, calculate the fraction: \(135 \div 180 = \frac{3}{4}\). Thus, our expression becomes \(\frac{3}{4} \times \pi\). This simplifies to \(\frac{3\pi}{4}\).
04

State the Answer in Radians

After simplifying, we find that \(135^{\circ}\) is equivalent to \(\frac{3\pi}{4}\) radians.

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Key Concepts

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

Degree to Radian Conversion
Converting degrees to radians is an essential concept in trigonometry and is necessary for handling angles in various mathematical computations. The heart of this process lies in the conversion formula:\[ \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \]This formula comes from the fundamental fact that a complete circle in radians, i.e.,
  • 360掳 equals 2蟺 radians,
  • and therefore 180掳 equals 蟺 radians.
Therefore, each degree is equivalent to \( \frac{\pi}{180} \) radians, making this fraction the conversion factor. When applying this factor to any angle measured in degrees, it allows you to re-express that angle in terms of radians.
Angle Conversion
Converting angles is a necessary skill for translating problems expressed in degrees to radians, which are often more useful in calculus and many areas of physics.
The conversion process is simple if you follow these steps:1. **Write down the degree measure** of the angle you want to convert. * In our exercise, it is 135掳.2. **Use the conversion formula**: multiply the degree measure by \( \frac{\pi}{180} \). * For example, \( 135^{\circ} \times \frac{\pi}{180} \).3. **Simplify the result** to get the radian measure.Through this method, you reduce computational errors. Therefore, it's important to become familiar with working through these conversions smoothly.
Trigonometry
Trigonometry, the study of angles and their relationships, hinges upon the understanding of radian measures.
Radians provide a natural connection between the angle and the arc length on a unit circle. This relationship is crucial when dealing with trigonometric functions such as sine, cosine, and tangent.
  • Radians are the preferred unit of measure in many mathematical contexts because of their direct relation to the radius of a circle.
  • Knowing how to convert degrees to radians enhances your ability to solve problems involving periodic functions and helps to understand wave patterns.
Understanding and utilizing trigonometry effectively requires fluency with angle conversions and familiarity with radian measure. Ultimately, these conversions make it easier to work within the broader scope of mathematics and sciences.

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Most popular questions from this chapter

Find the exact function values, if possible. Do not use your GDC. a) \(\cos \frac{5 \pi}{6}\) b) \(\sin 315^{\circ}\) c) \(\tan \frac{3 \pi}{2}\) d) \(\sec \frac{5 \pi}{3}\) e) \(\csc 240^{\circ}\)

Prove each identity.$$\frac{1}{\sec \theta(1-\sin \theta)}=\sec \theta+\tan \theta$$

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