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Chapter 8: Problem 19

determine the quadrant in which \(\theta\) lies.. $$ \sin \theta<0, \cos \theta>0 $$

Short Answer

Expert verified
\(\theta\) lies in Quadrant IV.

Step by step solution

01

Identify the Quadrants

First, we need to identify the four quadrants of a Cartesian plane. Quadrant I is where both \(y(\sin\theta)>0\) and \(x(\cos\theta)>0\), Quadrant II is where \(y(\sin\theta)>0\) but \(x(\cos\theta)<0\), Quadrant III is where both \(y(\sin\theta)<0\) and \(x(\cos\theta)<0\), and Quadrant IV is where \(y(\sin\theta)<0\) but \(x(\cos\theta)>0\).
02

Determine the Quadrant of \(\theta\)

We are given that \(\sin\theta<0\) which means \(y<0\), and \(\cos\theta>0\) which means \(x>0\). Thus \(\theta\) must lie in the quadrant where \(y<0\) and \(x>0\). From step 1, this quadrant is Quadrant IV.

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Key Concepts

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

Quadrants
The concept of quadrants is essential when discussing angles and trigonometric functions. Picture a Cartesian plane divided into four sections, each called a quadrant. The x-axis and y-axis intersect to form these quarters, allowing us to understand how sine and cosine values behave in different sections of the plane.

  • Quadrant I: Both sine (y) and cosine (x) values are positive.
  • Quadrant II: Sine is positive, cosine is negative.
  • Quadrant III: Both sine and cosine are negative.
  • Quadrant IV: Sine is negative, cosine is positive.
When determining the quadrant of an angle, it helps if you remember the behavior of sine (\( heta\)) and cosine (\( heta\)), each representing positions on the y-axis and x-axis, respectively.
Sine and Cosine
Sine and cosine are fundamental trigonometric functions that help describe the relationships between angles and ratios in a right triangle. In a unit circle, sine and cosine are used to project the angle's value onto the y-axis and x-axis, respectively. This projection helps determine the signs and thus the values of these functions in different quadrants.

  • Sine: This function describes the vertical position or y-coordinate. Its value indicates how far up or down from the origin an angle lies in the unit circle.
  • Cosine: This function describes the horizontal position or x-coordinate, telling how far left or right from the origin.
In terms of the problem, knowing that \( heta\) has \( ext{sin}\ heta < 0\) and \( ext{cos}\ heta > 0\) helps us determine its quadrant by reflecting these projections on the Cartesian plane.
Cartesian Plane
A Cartesian plane is a two-dimensional grid formed by the intersection of the x-axis (horizontal line) and y-axis (vertical line). This plane allows for the graphical representation of mathematical functions and coordinates, helping us to visualize their behavior and relations.

Key aspects of the Cartesian plane include:
  • Origin: The point where the x-axis and y-axis intersect, represented as (0,0).
  • Axes: The x-axis is generally horizontal, while the y-axis is vertical.
  • Quadrants: Divided by the axes into four sections, each designed to contain specific sign combinations of coordinates.
Understanding the Cartesian plane aids in analyzing angles and functions, as it provides a visual context for the behavior of sine and cosine, especially when determining the quadrant in which a particular angle lies.

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Most popular questions from this chapter

find the period and amplitude. $$ y=\frac{2}{3} \cos \frac{\pi x}{10} $$

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Meteorology The average monthly precipitation \(P\) in inches, including rain, snow, and ice, for Bismarck, North Dakota can be modeled by \(P=1.07 \sin (0.59 t+3.94)+1.52, \quad 0 \leq t \leq 12\) where \(t\) is the time in months, with \(t=1\) corresponding to January. $$ \begin{array}{l}{\text { (a) Find the maximum and minimum precipitation and the }} \\ {\text { month in which each occurs. }} \\ {\text { (b) Determine the average monthly precipitation for the }} \\ {\text { year. }} \\ {\text { (c) Find the total annual precipitation for Bismarck. }}\end{array} $$
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