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Chapter 5: Problem 13

$$ \text { If } 270^{\circ}

Short Answer

Expert verified
\(\cos(A/2)\) is negative.

Step by step solution

01

Understanding the Arc

The problem specifies that the angle \(A\) is in the range \(270^{\circ}<A<360^{\circ}\). This indicates that the angle \(A\) is in the fourth quadrant of the unit circle.
02

Half-Angle Range

Since \(A\) is between \(270^{\circ}\) and \(360^{\circ}\), the half of \(A\), which is \(\frac{A}{2}\), will be between \(135^{\circ}\) and \(180^{\circ}\) (as half the angle will halve the bounds of the angle). This puts \(\frac{A}{2}\) in the second quadrant of the unit circle.
03

Cosine in the Second Quadrant

Knowing that \(\frac{A}{2}\) is in the second quadrant is crucial because, in this quadrant, the cosine of any angle is negative. Therefore, \(\cos\left(\frac{A}{2}\right)\) will be negative.

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Key Concepts

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

Unit Circle
The unit circle is a fundamental concept in trigonometry that represents all possible angle measures in a plane. It is a circle with a radius of 1, centered at the origin (0,0) in the coordinate plane.
  • Angles are measured starting from the positive x-axis and rotating counterclockwise around the circle.
  • Each point on the circle relates to an angle and corresponds to coordinates \((\cos(\theta), \sin(\theta))\).
  • This method visualizes trigonometric functions, where the x-coordinate is the cosine of the angle, and the y-coordinate is the sine.
This circle helps students understand trigonometric identities since every angle has unique cosine and sine values. It provides a visual aid to determine the sign of these functions
depending on which quadrant the angle falls in.
Quadrants
Quadrants refer to the four sections of the coordinate plane divided by the x and y axes. Starting from the positive x-axis, the quadrants are numbered as follows:
  • First Quadrant: \(0^\circ \text{ to } 90^\circ\) - Cosine and sine values are both positive.
  • Second Quadrant: \(90^\circ \text{ to } 180^\circ\) - Sine is positive, cosine is negative.
  • Third Quadrant: \(180^\circ \text{ to } 270^\circ\) - Cosine and sine values are both negative.
  • Fourth Quadrant: \(270^\circ \text{ to } 360^\circ\) - Cosine is positive, sine is negative.
Knowing which quadrant an angle is located in helps determine the sign of its trigonometric functions. For instance, since the problem defines \(A\) between \(270^\circ\) and \(360^\circ\), it lies in the fourth quadrant,
where \(\cos\) is positive, making it crucial to determine the behavior of \(\cos(A/2)\) in the next step.
Half-Angle Formulas
Half-angle formulas are instrumental for calculating the trigonometric values of half an angle. They help express trigonometric functions of half angles in terms of the original angle:
  • For cosine, the half-angle formula is \[ \cos\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1+\cos A}{2}} \]
  • This formula's positive or negative outcome depends directly on the angle's position in the unit circle.
When \(A = 270^\circ < A < 360^\circ\), then dividing by 2 gives \(\frac{A}{2}\) between \(135^\circ\) and \(180^\circ\), meaning the angle \(\frac{A}{2}\)is in the second quadrant.
In this quadrant, \(\cos\left(\frac{A}{2}\right)\) is always negative, as determined by the characteristics inherent to this quadrant.

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Most popular questions from this chapter

The problems that follow review material we covered in Sections \(4.3\) and 4.6. Graph each of the following from \(x=0\) to \(x=4 \pi\). $$ y=2-2 \cos x $$

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