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Chapter 3: Problem 98

Describe how \(\cos t\) varies as \(t\) increases from \(\pi / 2\) to \(\pi\).

Short Answer

Expert verified
\( \cos t \) decreases from 0 to -1 as \( t \) increases from \( \pi/2 \) to \( \pi \).

Step by step solution

01

Understanding the range

First, let's understand the range given: from \( \pi/2 \) to \( \pi \). The unit circle is divided into four quadrants. From \( \pi/2 \) (90 degrees), you're starting at the top of the circle, moving left to \( \pi \) (180 degrees), which is the leftmost point on the circle.
02

Analyzing the cosine function

The cosine function, \( \cos t \), corresponds to the x-coordinate of a point on the unit circle. For angles between \( \pi/2 \) and \( \pi \), you're moving along the upper half of the unit circle from the top (x=0) to the leftmost point (x=-1).
03

Observing how cosine changes

As the angle \( t \) increases from \( \pi/2 \) to \( \pi \), the x-coordinate (which is \( \cos t \)) starts at 0 at \( \pi/2 \) and decreases to -1 at \( \pi \). Thus, \( \cos t \) is decreasing over this interval.

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Key Concepts

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

Understanding the Unit Circle
The unit circle is a fundamental tool in trigonometry, representing a circle with a radius of 1. It is centered at the origin of a coordinate plane, with its equation given as \( x^2 + y^2 = 1 \). This simple geometry helps us understand how trigonometric functions like sine and cosine derive their values.
  • At any given point on the unit circle, the x-coordinate and y-coordinate correspond to \( \cos t \) and \( \sin t \), respectively, based on the angle \( t \) formed with the positive x-axis.
  • The unit circle is divided into four quadrants, each associated with positive or negative values for sine and cosine. As the angle \( t \) changes, the corresponding point moves along the circle, altering both its x and y coordinates.
Understanding the unit circle allows us to see how the functions behave as they trace the different quadrants and helps visualize their cyclical nature.
Exploring the X-Coordinate on the Unit Circle
When exploring trigonometric functions through the unit circle, the x-coordinate of a point is crucial, as it determines the value of \( \cos t \). The x-coordinate indicates how far left or right the point is from the origin along the x-axis.
  • For angles framed in the first quadrant, the x-coordinate varies between \( 0 \) and \( 1 \), which means \( \cos t \) will take values in the positive range.
  • As angles reach into the second quadrant, between \( \pi/2 \) and \( \pi \), the x-coordinate swings to the left, decreasing from 0 to \(-1\). Here, \( \cos t \) becomes negative.
Understanding how the x-coordinate corresponds to \( \cos t \) is essential, especially when analyzing how its value changes as the angle progresses through different segments of the circle.
Grasping Angle Measurement in Radians
Angles can be measured in degrees or radians, with the latter being the standard unit used in many mathematical contexts.
  • One full rotation around the circle is equivalent to \( 360\) degrees or \( 2\pi \) radians. Thus, the angle \( \pi/2 \) corresponds to \( 90\) degrees, and \( \pi \) corresponds to \( 180\) degrees.
  • Measuring angles in radians offers a more direct connection to the unit circle. It simplifies complex equations and is integral when dealing with periodic functions.
Having a solid understanding of radians is crucial for interpreting trigonometric functions, as they relate directly to the arc length on the unit circle, making them invaluable for advanced mathematical applications.

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Most popular questions from this chapter

For Problems 83 through 94 , determine if the statement is possible for some real number \(z\). $$ \sec \frac{\pi}{2}=z $$

Use a calculator to approximate each value to four decimal places. $$ \sin 4 $$

Graph the unit circle using parametric equations with your calculator set to degree mode. Use a scale of 5 . Trace the circle to find all values of \(t\) between \(0^{\circ}\) and \(360^{\circ}\) satisfying each of the following statements. $$ \sin t=-\cos t $$

A person standing \(5.2\) feet from a mirror notices that the angle of depression from his eyes to the bottom of the mirror is \(13^{\circ}\), while the angle of elevation to the top of the mirror is \(12^{\circ}\). Find the vertical dimension of the mirror.

The problems that follow review material we covered in Section \(2.3\). Problems 107 through 112 refer to right triangle \(A B C\) in which \(C=90^{\circ}\). Solve each triangle. $$ A=42^{\circ}, c=36 $$
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