Problem 15 $f(x)=\cos x$, at \(x_{0}=\pi ... [FREE SOLUTION]

91影视

Differential Equations with Boundary Value Problems

John Polking, Al Boggess, David Arnold

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 1: Problem 15

$f(x)=\cos x$, at $x_{0}=\pi / 4$

Short Answer

Expert verified

$f(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.

Step by step solution

Understand the Problem

We are given a function $f(x) = \cos x$ and asked to evaluate or compute something specific at $x_0 = \frac{\pi}{4}$. We aim to find $f(x_0)$.

Substituting $x_0$ into the Function

Replace $x$ with $x_0$ in the function $f(x)$. Specifically, compute $f(\frac{\pi}{4}) = \cos(\frac{\pi}{4})$.

Evaluate the Trigonometric Function

Calculate $\cos(\frac{\pi}{4})$. Recall that $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.

Conclusion

The value of the function $f(x) = \cos x$ at $x_0 = \frac{\pi}{4}$ is $\frac{\sqrt{2}}{2}$.

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.

Function Evaluation

Function evaluation is the process of calculating a function's value for a specific input. It's like using a recipe: given certain ingredients, the function tells us the final result. Here, the function is given by $f(x) = \cos x$. To evaluate this function at a specific point, we substitute the given value into the function. For example, if we need to find $f$ at $x = \pi/4$, we replace $x$ in the function with $\pi/4$.
This involves substituting and simplifying trigonometric expressions, which can be straightforward but requires careful attention to detail. Let's explore the specifics of the cosine function next, which is the core part of our function evaluation here.

Cosine Function

The cosine function, denoted as $\cos(\theta)$, is a fundamental trigonometric function. It is defined in terms of the coordinates of an angle $\theta$ on the unit circle.
For any angle $\theta$, $\cos(\theta)$ gives the x-coordinate of the point on the unit circle corresponding to that angle.

The function has a range of [-1, 1], meaning its outputs are always between -1 and 1.
It is periodic with a period of $2\pi$, which means the function repeats its values every $2\pi$ units.
The cosine function is essential in many fields, such as physics and engineering, because it models wave-like phenomena.

In our example, we evaluated $\cos(\pi/4)$, an angle that resides in the first quadrant of the unit circle.

Value at Specific Point

Calculating the value of a trigonometric function at a specific point involves substituting the point into the function and simplifying the result. In our case, the point was $x_0=\pi/4$.
When we substitute $\pi/4$ into $f(x) = \cos x$, we get $\cos(\pi/4)$. This specific point belongs to the family of key angles, often memorized due to their frequent appearance in problems.

The value of $\cos(\pi/4)$ is $\frac{\sqrt{2}}{2}$, which comes from the geometric property of a 45-degree angle in a unit circle.
This angle leads to an isosceles right triangle where both legs are equal, giving us this result.

Recognizing these common values helps you quickly find solutions and deepen your understanding of trigonometric functions.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

\(f(x)=\cos x\), at \(x_{0}=\pi / 4\)

Short Answer

Step by step solution

Understand the Problem

Substituting \(x_0\) into the Function

Evaluate the Trigonometric Function

Conclusion

Key Concepts

Function Evaluation

Cosine Function

Value at Specific Point

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.