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Chapter 5: Problem 107

Find two solutions of each equation. Give your solutions in both degrees \(\left(0^{\circ} \leq \theta < 360^{\circ}\right)\) and radians \((0 \leq \theta < 2 \pi) .\) Do not use a calculator. (a) \(\sin \theta=\frac{1}{2}\) (b) \(\sin \theta=-\frac{1}{2}\)

Short Answer

Expert verified
The solutions for \(\sin \theta = \frac{1}{2}\) are: (饾渻=30掳 or 饾渻=150掳) and (饾渻=蟺/6 or 饾渻=5蟺/6). The solutions for \(\sin \theta = -\frac{1}{2}\) are: (饾渻=210掳 or 饾渻=330掳) and (饾渻=7蟺/6 or 饾渻=11蟺/6).

Step by step solution

01

Identify Sine Values on the Unit Circle

On the unit circle, \(\sin \theta=\frac{1}{2}\) when \(\theta\) equals 30掳 or 150掳. Convert these angles to radians: \(30^\circ=\frac{\pi}{6}\) rad and \(150^\circ=\frac{5\pi}{6}\) rad.
02

Find Negative Sine Values on the Unit Circle

The sine function is negative in the third and fourth quadrants of the unit circle. So, \(\sin \theta=-\frac{1}{2}\) when \(\theta\) equals 210掳 or 330掳. Convert these angles to radians: \(210^\circ=\frac{7\pi}{6}\) rad and \(330^\circ=\frac{11\pi}{6}\) rad.

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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 fundamental in trigonometry, providing a simple framework for understanding angles and their sine values. It's a circle with a radius of 1, centered at the origin of a coordinate plane. Any point on the unit circle can be described using coordinates \( (\cos \theta, \sin \theta) \).
This connection means every angle has a corresponding sine value located at the vertical axis. Here's how it breaks down:
  • For angles between 0掳 and 90掳 (first quadrant), sine values are positive.
  • For angles between 90掳 and 180掳 (second quadrant), sine values remain positive.
  • Between 180掳 and 270掳 (third quadrant), sine values are negative.
  • Finally, for angles from 270掳 to 360掳 (fourth quadrant), sine values stay negative.
The unit circle helps us find trigonometric values without a calculator, by remembering specific angles that are often utilized, like 30掳, 45掳, and 60掳, among others.
Sine Function
The sine function is a periodic function that relates the angle of a right triangle to the ratio of the length of the opposite side over the hypotenuse. On the unit circle, the sine of angle \(\theta\) is the y-coordinate of the corresponding point.
The key features include:
  • The sine function has a periodicity of \(2\pi\), meaning it repeats every \(360^{\circ}\) or \(2\pi\) radians.
  • It has an amplitude of 1, as it ranges between -1 and 1 due to the unit circle's radius.
  • Sine is positive in the first and second quadrants and negative in the third and fourth quadrants on the unit circle.
From the exercise, we observe \(\sin \theta = \frac{1}{2}\) occurs at 30掳 and 150掳, and in radians, these are \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\). Similarly, \(\sin \theta = -\frac{1}{2}\) in the third and fourth quadrants is found at angles 210掳 and 330掳, converting to radians as \(\frac{7\pi}{6}\) and \(\frac{11\pi}{6}\).
Degrees to Radians Conversion
Converting between degrees and radians is crucial in trigonometry, making it easier to apply mathematical principles across different problems. The relationship between degrees and radians is based on the circle's total rotation: 360掳 equals \(2\pi\) radians.
Here's a simple conversion guide:
  • To convert degrees to radians, use the formula \(\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\).
  • For example, 30掳 converts to radians as \(30 \times \frac{\pi}{180} = \frac{\pi}{6}\).
  • Similarly, 150掳 converts to radians as \(150 \times \frac{\pi}{180} = \frac{5\pi}{6}\).
  • The reverse can also be done by multiplying radians by \(\frac{180}{\pi}\) to get degrees.
Using these conversions, you can easily interchange between degrees and radians, making trigonometric calculations more intuitive and adaptable.

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