/*! 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 76 Determine if the statement is po... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 76

Determine if the statement is possible for some real number \(z\). \(\sin 0=z\)

Short Answer

Expert verified
The statement is possible for some real number \(z\). Specifically, \(z = 0\).

Step by step solution

01

Understand the range of the sine function

Recall that the sine function \(sin(\theta)\) can take values between -1 and 1 for any real number \(\theta\). This means that for any angle \(\theta\), the output of \(\sin(\theta)\) cannot exceed this range.
02

Evaluate \\(\sin 0\\)

Substitute \(0\) into the sine function to find its value, \(\sin(0)\). We know from the unit circle that \(\sin(0) = 0\).
03

Compare \\(\sin 0\\) with \\(z\\)

The statement in question is \(\sin 0 = z\). Since from Step 2 we determined that \(\sin(0) = 0\), it follows that this statement is true if \(z = 0\).
04

Determine if the statement is possible

The statement \(\sin 0 = z\) can be true for \(z = 0\). Since 0 is a real number, the statement is possible.

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.

Unit Circle
When dealing with trigonometric functions like sine, the unit circle is a fundamental concept that helps us understand these functions. The unit circle is a circle centered at the origin of a coordinate plane with a radius of exactly one unit. Every point on this circle corresponds to an angle measured from the positive x-axis.
  • The coordinates of each point on the unit circle are in the form \( (\cos(\theta), \sin(\theta)) \).
  • The angle \(\theta\) is measured in radians, a unit of angular measure used in mathematics.
  • Using the unit circle, geometrically, means that the y-coordinate of any point on the circle gives you the value of the sine of that angle.
By recognizing this pattern, you can easily find the sine of standard angles such as \(0, \pi/6, \pi/4, \pi/3, \) or \(\pi/2\). For example, at an angle of \(0\), the corresponding point on the unit circle is \((1, 0)\), thus the sine of \(0\) is exactly 0.
Range of Sine Function
The sine function is one of the primary trigonometric functions, and it possesses a unique range. Understanding the range is crucial for solving many trigonometric equations. The range of the sine function is the set of all possible values that the function can take.
  • The sine wave moves up and down in a regular pattern, creating a smooth curve.
  • Mathematically, the sine function only takes values from -1 to 1, inclusive.
  • This means for any angle \(\theta\), \(-1 \leq \sin(\theta) \leq 1\).
Using this knowledge, if you encounter an equation involving sine, such as \(\sin(\theta) = z\), you can immediately know that \(z\) must fall within this interval. It simplifies the process of determining what is possible for the sine function.
Evaluating Trigonometric Functions
Evaluating trigonometric functions, like the sine function, involves calculating the value of the function at a specific angle. This process is straightforward once you are familiar with the unit circle and understand the function's range.
  • The angle can be in degrees or radians; it's crucial to know which unit is being used.
  • For example, evaluating \(\sin(0)\) involves checking the y-coordinate in the unit circle at angle \(0\), which is clearly \(0\).
  • For more complicated angles, you might need to use trigonometric identities or a calculator.
By mastering these basics, you can confidently find the value of sine and other trigonometric functions at any given angle. Developing a solid understanding of these steps is key to tackling more complex trigonometric problems easily.

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

When Lance Armstrong blazed up Mount Ventoux in the 2002 Tour, he was equipped with a 150 -millimeter-diameter chainring and a 95 -millimeterdiameter sprocket. Lance is known for maintaining a very high cadence, or pedal rate. If he was pedaling at a rate of 90 revolutions per minute, find his speed in kilometers per hour. \((1 \mathrm{~km}=1,000,000 \mathrm{~mm})\)

Draw each of the following angles in standard position and then name the reference angle. $$ 210^{\circ} $$

Distance and Bearing A ship sails for \(85.5\) miles on a bearing of \(\mathrm{S} 57.3^{\circ} \mathrm{W}\). How far west and how far south has the boat traveled?

Use a calculator to find \(\theta\) to the nearest tenth of a degree, if \(0^{\circ}<\theta<360^{\circ}\) and $$ \tan \theta=0.5890 \text { with } \theta \text { in OIII } $$

Find \(\theta\), if \(0^{\circ}<\theta<360^{\circ}\) and $$ \sec \theta=\sqrt{2} \text { with } \theta \text { in QIV } $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Decision 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.