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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) -1/2, (b) \(\sqrt{2}/2\)

Step by step solution

01

Determine the Reference Angle for 210掳

The angle 210掳 is in the third quadrant where sine is negative. The reference angle is the positive acute angle that 210掳 makes with the x-axis. To find it, subtract 180掳 from 210掳: \[ 210^{\circ} - 180^{\circ} = 30^{\circ} \] Thus, the reference angle is 30掳.
02

Calculate Sine of the Reference Angle

The sine value for the reference angle of 30掳 is known from the unit circle: \[ \sin 30^{\circ} = \frac{1}{2} \] Since 210掳 is in the third quadrant, where sine is negative, \[ \sin 210^{\circ} = -\sin 30^{\circ} = -\frac{1}{2} \]
03

Determine the Reference Angle for -315掳

Convert -315掳 to a positive angle by adding 360掳: \[ -315^{\circ} + 360^{\circ} = 45^{\circ} \] This means -315掳 has the same trigonometric values as 45掳.
04

Calculate Sine of the Positive Angle

Since -315掳's equivalent positive angle is 45掳, and knowing that sine is positive in the first quadrant where 45掳 lies:\[ \sin 45^{\circ} = \frac{\sqrt{2}}{2} \] Therefore, \[ \sin (-315^{\circ}) = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \]

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Key Concepts

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

Sine Function
The sine function is a fundamental concept in trigonometry, representing the ratio of the length of the side opposite to a given angle in a right triangle to the length of the hypotenuse. More formally, if you have a right triangle with an angle \( \theta \), the sine of \( \theta \) can be expressed as: \[ \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \] This function is periodic and has a range from -1 to 1, meaning that the sine of any angle will always result in a value between -1 and 1.
  • It repeats every 360掳, which means adding or subtracting 360掳 from an angle provides the same sine value.
  • It's useful in problems involving waveforms and oscillations owing to its periodic nature.
Understanding the sine function is crucial because it helps in deriving other trigonometric functions and solving real-world problems involving angles and lengths.
Reference Angle
The concept of a reference angle is a handy tool in trigonometry. A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis.
  • It simplifies the process of finding sine, cosine, and tangent values for angles larger than 90掳 or negative angles.
  • The reference angle is always a positive acute angle (less than 90掳), which simplifies calculations.
For example, in finding the reference angle for 210掳, you subtract 180掳 from 210掳, arriving at 30掳. This 30掳 is the reference angle and helps in calculating trigonometric values since standard angle values on the unit circle are easier to work with. Similarly, for an angle like -315掳, you convert it to a positive angle equivalent by adding 360掳, obtaining 45掳, then applying calculations using this reference angle.
Unit Circle
The unit circle is a circle with a radius of one, centered at the origin (0,0) on the coordinate plane. It provides a simple way to visualize trigonometric functions like sine and cosine.
  • Each point on the unit circle corresponds to the cosine and sine of an angle.
  • For a given angle \( \theta \), the x-coordinate of the point on the unit circle is \( \cos(\theta) \) and the y-coordinate is \( \sin(\theta) \).
The unit circle is particularly powerful because the sine and cosine of angles are repeatable and symmetrical, meaning you can use these properties to solve problems for angles greater than 360掳 or negative angles. Understanding the basics of the unit circle allows for easier comprehension of trigonometric identities and relationships, like finding sine and cosine values for reference angles obtained.
Quadrants
In the coordinate system, quadrants are the four sections of the plane, split by the x-axis and y-axis. Each quadrant has distinct properties regarding the signs of trigonometric functions.
  • The first quadrant: both sine and cosine are positive here.
  • The second quadrant: sine is positive, but cosine is negative.
  • The third quadrant: both sine and cosine are negative.
  • The fourth quadrant: sine is negative, but cosine is positive.
Recognizing which quadrant an angle resides in helps determine the signs of trigonometric values. For 210掳, which sits in the third quadrant, sine values are negative. In contrast, -315掳 converts to 45掳, placing it in the first quadrant where sine remains positive. Understanding these quadrant properties is essential for correctly determining the value of trigonometric functions and solving related problems.

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