/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 \(\sin \left(\frac{\pi}{4}\right... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 2

\(\sin \left(\frac{\pi}{4}\right)= ______\quad ; \cos \left(\frac{8 \pi}{3}\right)= _____ \)

Short Answer

Expert verified
\[\text{sin}(\frac{\text{蟺}}{4}) = \frac{\text{鈭�2}}{2}\], \[\text{cos}(\frac{8\text{蟺}}{3}) = -\frac{1}{2}\]

Step by step solution

01

Evaluate \(\text{sin}(\frac{\text{蟺}}{4})\)

Recall that \(\frac{\text{蟺}}{4}\) radians is equivalent to 45 degrees. The sine of 45 degrees is a known value: \(\text{sin}(45^\text{o}) = \frac{\text{鈭�2}}{2}\). Thus, \(\text{sin}(\frac{\text{蟺}}{4}) = \frac{\text{鈭�2}}{2}\).
02

Simplify \(\text{cos}(\frac{8\text{蟺}}{3})\)

First, simplify the angle \(\frac{8\text{蟺}}{3}\) by reducing it within the range of \([0, 2\text{蟺}]\). \(\frac{8\text{蟺}}{3}\) is equivalent to \(\frac{8\text{蟺}}{3} = 2\text{蟺} + \frac{2\text{蟺}}{3}\). Since cosine is periodic with period \2\text{蟺}\, we can reduce this to \(\text{cos}(\frac{2\text{蟺}}{3})\).
03

Evaluate \(\text{cos}(\frac{2\text{蟺}}{3})\)

Recall that \(\frac{2\text{蟺}}{3}\) radians is equivalent to 120 degrees. The cosine of 120 degrees is a known value: \(\text{cos}(120^\text{o}) = -\frac{1}{2}\). Thus, \(\text{cos}(\frac{2\text{蟺}}{3}) = -\frac{1}{2}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Sine Function
The sine function is one of the primary trigonometric functions. It relates an angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. In mathematical terms, for an angle \theta, the sine function is represented as: \(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).

For example, consider \(\text{sin}\big(\frac{\text{蟺}}{4}\big)\). The angle \(\frac{\text{蟺}}{4}\) or 45 degrees gives us \(\text{sin}(45^\text{o}) = \frac{\text{鈭�2}}{2}\). This value is derived from the unit circle or right triangle properties.
Cosine Function
The cosine function is another fundamental trigonometric function. It relates the length of the adjacent side of an angle to the hypotenuse in a right triangle. For an angle \theta, it is expressed as: \(\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).

For instance, the problem includes evaluating \(\text{cos}(\frac{8\text{蟺}}{3})\). This can be simplified because the cosine function is periodic. Only angles within the range \([0, 2\text{蟺}]\) need considering. Here, \(\frac{8\text{蟺}}{3}\) is equivalent to \(2\text{蟺} + \frac{2\text{蟺}}{3}\). Given the periodicity, \(\text{cos}\big(\frac{2\text{蟺}}{3}\big) = -\frac{1}{2}\).
Radians to Degrees Conversion
Understanding radians and their conversion to degrees is crucial in trigonometry. Radians are another measure for angles used mainly in higher mathematics. To convert radians to degrees, use the conversion: \(1\text{ radian} = \frac{180^\text{o}}{\text{蟺}}\).

For example, converting \(\frac{\text{蟺}}{4}\) to degrees: \(\frac{\text{蟺}}{4} \times \frac{180^\text{o}}{\text{蟺}} = 45^\text{o}\). Similarly, to understand \(\frac{2\text{蟺}}{3}\) in degrees, use: \(\frac{2\text{蟺}}{3} \times \frac{180^\text{o}}{\text{蟺}} = 120^\text{o}\). This simplification helps evaluate trigonometric functions more easily.
Periodic Functions
Trigonometric functions like sine and cosine are periodic. This means they repeat their values over regular intervals. Specifically, sine and cosine have a period of \(2\text{蟺}\). This periodicity implies \(\text{sin}(\theta) = \text{sin}(\theta + 2\text{蟺}k)\) and \(\text{cos}(\theta) = \text{cos}(\theta + 2\text{蟺}k)\) for any integer k.

In the given problem, \(\frac{8\text{蟺}}{3}\) can be simplified using this property to \(\frac{8\text{蟺}}{3} = 2\text{蟺} + \frac{2\text{蟺}}{3}\). Recognizing such transformations makes solving trigonometric problems straightforward.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write as a single logarithm: $$ 3 \log _{7} x+2 \log _{7} y-5 \log _{7} z $$

Show that \(\sin \left(\sin ^{-1} v+\cos ^{-1} v\right)=1\)

Show that \(\cos \left(\sin ^{-1} v+\cos ^{-1} v\right)=0\)

The speed of yellow sodium light (wavelength, 589 nanometers) in a certain liquid is measured to be \(1.92 \times 10^{8}\) meters per second. What is the index of refraction of this liquid, with respect to air, for sodium light?

If \(f(x)=\frac{8}{x^{2}},\) find an equation of the secant line containing the points \((1, f(1))\) and \((4, f(4))\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.