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Chapter 5: Problem 8

Find the exact value. (a) \(\sin 210^{\circ}\) (b) \(\sin \left(-315^{\circ}\right)\)

Short Answer

Expert verified
(a) -\(\frac{1}{2}\) and (b) \(\frac{\sqrt{2}}{2}\).

Step by step solution

01

Determine the Reference Angle for Part (a)

The angle 210掳 is in the third quadrant. First, find the reference angle by subtracting 180掳 from 210掳. So, Reference Angle = 210掳 - 180掳 = 30掳.
02

Calculate \\(\sin 210^{\circ}\\)

In the third quadrant, the sine function is negative. Use the reference angle 30掳 and the identity that \(\sin(210^{\circ}) = -\sin(30^{\circ})\).Since \(\sin(30^{\circ}) = \frac{1}{2}\), we have \(\sin(210^{\circ}) = -\frac{1}{2}\).
03

Determine the Reference Angle for Part (b)

The angle \(-315^{\circ}\) is a negative angle, which mean we rotate clockwise. Convert it to a positive angle by adding 360掳.So, \(-315^{\circ} + 360^{\circ} = 45^{\circ}\).
04

Calculate \\(\sin(-315^{\circ})\\)

The angle 45掳 obtained from the previous step is in the first quadrant. Here, sine is positive.Thus, \(\sin(-315^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\).

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Key Concepts

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

Understanding Reference Angles
Reference angles are a crucial part of trigonometry. They help in determining the trigonometric values for non-standard angles. A reference angle is the smallest angle between the terminal side of a given angle and the horizontal axis. This angle is always positive and between 0掳 and 90掳. For an angle in any quadrant, you can find the reference angle by using specific rules based on the quadrant. For angles in the first quadrant, the reference angle is the angle itself. In the second quadrant, subtract the angle from 180掳. In the third quadrant, subtract 180掳 from the angle. Lastly, in the fourth quadrant, subtract the angle from 360掳.
Exploring Quadrants in Trigonometry
In trigonometry, the coordinate system is divided into four sections known as quadrants. These act as a canvas for angles and help determine the sign of trigonometric functions.
  • First Quadrant: All trigonometric functions are positive.
  • Second Quadrant: Sine is positive, but cosine and tangent are negative.
  • Third Quadrant: Tangent is positive, while sine and cosine are negative.
  • Fourth Quadrant: Cosine is positive, whereas sine and tangent are negative.
Recognizing in which quadrant an angle lies helps resolve whether the function value is positive or negative.
Delving into the Sine Function
The sine function is a fundamental component of trigonometry. It maps the angle measure to a value between -1 and 1. Sine represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. It is periodic with a period of 360掳, meaning the values repeat after every 360掳. To evaluate the sine of an angle, it's essential to understand its behavior in different quadrants. For example, in the third quadrant (as in the case of 210掳), the sine value is negative.
Working with Negative Angles
Negative angles might seem tricky at first, but they follow straightforward rules. In trigonometry, a negative angle indicates a clockwise rotation from the positive x-axis. To convert a negative angle to a positive one, you can add 360掳 until you get a positive equivalent angle. As seen with the angle (-315掳 + 360掳 = 45掳), this adjustment makes it easier to use standard trigonometric principles. By converting negative angles to their positive counterparts, it simplifies finding exact trigonometric values.
Determining Exact Trigonometric Values
Exact trigonometric values are evaluated without any approximation, often using known angle values. Certain angles like 30掳, 45掳, and 60掳 have well-known sine, cosine, and tangent values that frequently occur in trigonometry. For instance, \(\sin(30掳) = \frac{1}{2}\)and \(\sin(45掳) = \frac{\sqrt{2}}{2}\).These values are vital as they are frequently applied when using reference angles to resolve more complex angles inside different quadrants.Acquiring these exact values equips you with the ability to solve various trigonometric problems efficiently.

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Most popular questions from this chapter

Pizza values A vender sells two sizes of pizza by the slice. The small slice is \(\frac{1}{6}\) of a circular 18 -inch-diameter pizza, and it sells for \(\$ 2.00 .\) The large slice is \(\frac{1}{8}\) of a circular 26 -inchdiameter pizza, and it sells for \(\$ 3.00 .\) Which slice provides more pizza per dollar?

Potation of compact disces (COs) The drive motor of a particular CD player is controlled to rotate at a speed of \(200 \mathrm{rpm}\) when reading a track 5.7 centimeters from the center of the CD. The speed of the drive motor must vary so that the reading of the data occurs at a constant rate. (a) Find the angular speed (in radians per minute) of the drive motor when it is reading a track 5.7 centimeters from the center of the CD.(b) Find the linear speed (in \(\mathrm{cm} / \mathrm{sec}\) ) of a point on the CD that is 5.7 centimeters from the center of the CD. Find the angular speed (in rpm) of the drive motor when it is reading a track 3 centimeters from the center of the CD. Find a function \(S\) that gives the drive motor speed in \(\mathrm{rpm}\) for any radius \(r\) in centimeters, where \(2.3 \leq r \leq 5.9\) What type of variation exists between the drive motor speed and the radius of the track being read? Check your answer by graphing \(S\) and finding the speeds for \(r=3\) and \(r=5.7\)

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta .\) $$\sqrt{\cos ^{2}(\theta / 2)} ; \quad 0<\theta<\pi$$

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