/*! 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 19 \(f(x)=\sin 2 x\), at \(x=\pi / ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 19

\(f(x)=\sin 2 x\), at \(x=\pi / 8\)

Short Answer

Expert verified
Evaluate \(f\left(\frac{\pi}{8}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

Step by step solution

01

Identify the Function and Given Value

The function given is \(f(x) = \sin(2x)\), and we need to evaluate this function at \(x = \frac{\pi}{8}\).

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 fundamental trigonometric functions. It is typically written as \(\sin(x)\) and deals with the ratio of the length of the side opposite to the angle to the hypotenuse in a right triangle. This makes it an essential tool in not only geometry but also in various fields such as physics and engineering. The sine function has a periodic, wavelike structure with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) units. The fundamental wave starts at the origin (0,0), reaches a maximum value of 1, and a minimum of -1. Thus, the range of the sine function is always between -1 and 1.
Function Evaluation
Evaluating a function means substituting the given values into the function to find the result. In this exercise, we are asked to evaluate the function \(f(x) = \sin(2x)\) at \(x = \frac{\pi}{8}\). This involves replacing \(x\) in the equation with \(\frac{\pi}{8}\), transforming it into \(f\left(\frac{\pi}{8}\right) = \sin\left(2 \times \frac{\pi}{8}\right)\). Simplifying within the function by calculating \(2 \times \frac{\pi}{8} = \frac{\pi}{4}\) allows us to substitute into \(\sin\), resulting in \(f\left(\frac{\pi}{8}\right) = \sin\left(\frac{\pi}{4}\right)\). This process is a straightforward way to find the value of a function for specific inputs.
Angle Transformation
Angle transformation is essential when dealing with trigonometric functions. It involves manipulating the angle within the function to simplify the calculation process. In our exercise, we transform the angle within the sine function from the double angle \(2x\) to a simpler expression by using \(x = \frac{\pi}{8}\). Upon substitution, \(2x\) becomes \(\frac{\pi}{4}\), which is a standard angle whose sine value is often memorized or can be easily derived from the unit circle. The sine of \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\). Knowing how to transform and evaluate angles efficiently is crucial in simplifying the steps needed to solve trigonometric problems.
Mathematical Problem Solving
Mathematical problem solving involves clear logic and methodology to arrive at a solution. In solving the problem \(f(x)=\sin(2x)\) at \(x=\frac{\pi}{8}\), we applied a series of logical steps. We started by identifying the function and the given value of \(x\). We then conducted function evaluation by substituting \(x = \frac{\pi}{8}\) into the sine function, transforming the angle to another form, which in this case, simplifies to the sine of \(\frac{\pi}{4}\). Lastly, using our knowledge of the sine values of standard angles, we calculated the final value of the function. This methodical approach showcases the step-by-step work that embodies mathematical problem-solving.

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

\(f(x)=\sqrt{x+1}\), at \(x=0\)

\(y^{\prime}=\frac{3 t}{1+2 t^{2}}\)

A ball is dropped from rest from a height of \(200 \mathrm{~m}\). What is the velocity and position of the ball 3 seconds later?

A particle moves along the \(x\)-axis, its position from the origin at time \(t\) given by \(x(t)\). A single force acts on the particle that is proportional to, but opposite the object's displacement. Use Newton's law to derive a differential equation for the object's motion. X

\(y^{\prime}=\frac{t}{1+t^{2}}\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.