Problem 107 - Algebra and Trigonometry Real Mathematics, Real People

Question

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) \(\sin \theta=-\frac{1}{2}\)

Step-by-Step Solution

Answer

The solutions for \(\sin \theta = \frac{1}{2}\) are: (𝜃=30° or 𝜃=150°) and (𝜃=π/6 or 𝜃=5π/6). The solutions for \(\sin \theta = -\frac{1}{2}\) are: (𝜃=210° or 𝜃=330°) and (𝜃=7π/6 or 𝜃=11π/6).

Step 1: Identify Sine Values on the Unit Circle

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

Step 2: 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° or 330°. Convert these angles to radians: \(210^\circ=\frac{7\pi}{6}\) rad and \(330^\circ=\frac{11\pi}{6}\) rad.

Key Concepts

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° and 90° (first quadrant), sine values are positive.
• For angles between 90° and 180° (second quadrant), sine values remain positive.
• Between 180° and 270° (third quadrant), sine values are negative.
• Finally, for angles from 270° to 360° (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°, 45°, and 60°, 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° and 150°, 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° and 330°, 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° 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° converts to radians as \(30 \times \frac{\pi}{180} = \frac{\pi}{6}\).
• Similarly, 150° 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.