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Chapter 5: Problem 14

$$ \text { If } 180^{\circ}

Short Answer

Expert verified
The sine of \(A/2\) is positive because \(A/2\) lies in the second quadrant where sine is positive.

Step by step solution

01

Determine the Quadrant

Angles between 180掳 and 270掳 lie in the third quadrant of the unit circle. In this quadrant, sine values are negative while cosine values are also negative.
02

Identify the Properties of Half-Angle

The half-angle formula for sine is given by \( \sin\left( \frac{A}{2} \right) = \pm \sqrt{\frac{1-\cos(A)}{2}} \). Whether it is positive or negative depends on the quadrant of \( \frac{A}{2} \).
03

Determine the Quadrant for Half of Angle A

If \(180^{\circ} < A < 270^{\circ}\), then \(90^{\circ} < \frac{A}{2} < 135^{\circ}\). This range lies in the second quadrant of the unit circle.
04

Analyze the Sign of Sine in Quadrant 2

In the second quadrant, the sine function is positive. Therefore, \( \sin\left( \frac{A}{2} \right) \) is positive.

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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 crucial tool in trigonometry. It simplifies understanding the sine, cosine, and tangent functions as they relate to angles and triangles. A unit circle is a circle with a radius of 1, centered at the origin (0, 0) of a coordinate plane.
  • The x-coordinate of a point on the unit circle corresponds to the cosine of the angle.
  • The y-coordinate corresponds to the sine of the angle.
  • The angle, measured in radians, increases counterclockwise starting from the positive x-axis.
The circle divides the plane into four quadrants. Each quadrant has distinct characteristics for trigonometric functions:
  • First Quadrant (0掳 - 90掳): Both sine and cosine are positive.
  • Second Quadrant (90掳 - 180掳): Sine is positive, cosine is negative.
  • Third Quadrant (180掳 - 270掳): Both sine and cosine are negative.
  • Fourth Quadrant (270掳 - 360掳): Sine is negative, cosine is positive.
Understanding the unit circle helps in predicting the sign and value of trigonometric functions at different angles. For example, since the angle A is between 180掳 and 270掳, it falls in the third quadrant where the sine function is negative.
Sine Function
The Sine Function is a fundamental trigonometric function with a periodic wave form. It describes the height, or y-coordinate, of a point on the unit circle as a function of angle.
Important properties of the sine function include:
  • It has a periodicity of 360掳 or \(2\pi\) radians, repeating its values every full rotation around the circle.
  • The function oscillates between -1 and 1.
  • At 0掳 and 180掳, the sine is 0.
  • At 90掳, it reaches its peak value of 1, while at 270掳, it dips to -1.
When studying the sine of half an angle, it's important to understand in which quadrant the resulting half-angle lies. For the provided problem, if the full angle A is in the third quadrant, the half-angle \(\frac{A}{2}\) will be in the second quadrant where the sine function becomes positive.
This confirms that \(\sin\left(\frac{A}{2}\right)\) is positive, even though \(\sin(A)\) is negative in its native third quadrant.
Half-Angle Formulas
Half-Angle Formulas are trigonometric identities that allow the calculation of sine, cosine, and tangent for half of an angle. They are particularly handy when reducing complex trigonometric expressions:
For the sine function, the half-angle formula is:\[\sin\left( \frac{A}{2} \right) = \pm \sqrt{\frac{1-\cos(A)}{2}}\]The sign of the result depends on the quadrant in which the half-angle lies:
  • In the first and second quadrants, sine is positive.
  • In the third and fourth quadrants, sine is negative.
In our exercise, given that \(180^{\circ}

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