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Chapter 5: Problem 16

Determine the quadrant in which each angle lies. (a) \(87.9^{\circ}\) (b) \(-8.5^{\circ}\)

Short Answer

Expert verified
(a) \(87.9^{\circ}\) lies in Quadrant I. (b) \(-8.5^{\circ}\) lies in Quadrant IV.

Step by step solution

01

Identify the Quadrant for \(87.9^{\circ}\)

\(87.9^{\circ}\) falls between \(0^{\circ}\) and \(90^{\circ}\), so it lies in Quadrant I.
02

Identify the Quadrant for \(-8.5^{\circ}\)

\(-8.5^{\circ}\) is a negative angle, meaning it's measured clockwise from the positive x-axis. Since it's less than \(90^{\circ}\), it still lies in Quadrant IV.

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Key Concepts

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

Angle Measurement
Angles are a fundamental aspect of trigonometry and geometry. They define the rotation or the amount of turn between two rays originating from a common point called the vertex. These angles can be measured in degrees, where a full circle is equivalent to 360 degrees. This measurement system allows us to understand the position of different angles relative to a circular or a Cartesian coordinate system.

When considering degrees, it is important to see these as segments of a circle:
  • Each quadrant of the circle spans 90 degrees.
  • Hence, Quadrant I extends from 0 to 90 degrees, Quadrant II from 90 to 180 degrees, Quadrant III from 180 to 270 degrees, and Quadrant IV from 270 to 360 degrees.
Knowing this helps in determining where an angle lies within the coordinate system.
Negative Angles
Negative angles are essentially the complement of positive angles, measured in the opposite or clockwise direction. When dealing with negative angles, it is key to remember that the standard positive angle measurement moves counterclockwise from the positive x-axis.

For negative angles:
  • They move clockwise.
  • A circle's revolution remains at 360 degrees, so for a negative measurement like \(-8.5^{\circ}\), we can add 360 degrees to better understand its position, giving \(351.5^{\circ}\).
  • This conversion might help visualize them in the context of positive angle measurement as many find clockwise measurement less intuitive.
Understanding how negative angles fit in will empower you to determine which quadrant they fall into, just like positive angles, but mirrored.
Quadrant Determination
Determining the quadrant of an angle requires understanding of the coordinate plane division. The coordinate plane is divided into four quadrants:

  • Quadrant I: angles between 0 and 90 degrees
  • Quadrant II: angles between 90 and 180 degrees
  • Quadrant III: angles between 180 and 270 degrees
  • Quadrant IV: angles between 270 and 360 degrees
When you have any angle measure, figuring out the quadrant involves assessing this range.

Consider negative angles by adding 360 degrees if it helps revisualize them. For example, an angle of \(-8.5^{\circ}\) is equivalent to \(351.5^{\circ}\) in positive measurement, squarely placing it in Quadrant IV.

Thus, familiarizing yourself with these quadrant ranges and considering the clockwise nature of negative angles helps you swiftly determine the correct position of any given angle.

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Most popular questions from this chapter

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=60^{\circ}$$

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Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials $$\sin x \approx x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !} \quad \text { and } \quad \cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}$$ where \(x\) is in radians. (a) Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Use the graphing utility to graph the cosine function and its polynomial approximation in the same viewing window. How do the graphs compare? (c) Study the patterns in the polynomial approximations of the sine and cosine functions and predict the next term in each. Then repeat parts (a) and (b). How did the accuracy of the approximations change when an additional term was added?

Use a graphing utility to graph the logarithmic function. Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function. $$f(x)=\log _{3} x$$

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-5 \pi / 3$$
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