Problem 6 - Algebra 2

Question

Use a unit circle and \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles to find the degree measures of the angles. angles whose sine is \(-\frac{1}{2}\)

Step-by-Step Solution

Answer

The angles whose sine is \(-\frac{1}{2}\) are \(210^{\circ}, 330^{\circ}, -30^{\circ}\), and \(-150^{\circ}\).

Step 1: Identify Quadrants

As the sine of the angle is negative, which means the angle lies in the third or fourth quadrant.

Step 2: Recognize Special Angles

Recognize that the sine of an angle is \(-\frac{1}{2}\) at \(30^{\circ}\) or \(150^{\circ}\). Now, reflect these angles into the third and fourth quadrant.

Step 3: Calculate the Angles

For the third quadrant, add \(180^{\circ}\) to \(30^{\circ}\) and \(150^{\circ}\) which gives \(210^{\circ}\) and \(330^{\circ}\). Turning by more than one full circle, subtract \(360^{\circ}\) from both of these, which results in \(-150^{\circ}\) and \(-30^{\circ}\). For the fourth quadrant, subtract \(30^{\circ}\) and \(150^{\circ}\) from \(360^{\circ}\), which gives \(330^{\circ}\) and \(210^{\circ}\). Turning in the clockwise direction, change the signs, which gives \(-30^{\circ}\) and \(-150^{\circ}\).

Key Concepts

Understanding the Unit Circle

The unit circle is a critical concept in trigonometry and a valuable tool when working with angles and trigonometric functions. It is called the 'unit' circle because it has a radius of 1.
This simplicity allows us to easily see and understand the behavior of trigonometric functions. The circle is centered at the origin of the coordinate system, allowing angles to be measured in standard position, starting from the positive x-axis and moving counterclockwise.
The coordinates of any point on the circumference of the circle can be

• x-coordinate: represents the cosine of the angle
• y-coordinate: represents the sine of the angle

The circle divides the plane into four quadrants, and understanding which quadrant an angle is in helps determine the sign of the sine, cosine, and tangent functions. In the unit circle, sine values are positive in the first and second quadrants, while negative in the third and fourth quadrants.

Exploring Special Angles

Special angles, such as those found in a 30-60-90 triangle, are key in trigonometry for their consistent ratios and simple conversion to radians.
These angles are easy to recall and utilize because they stem from highly symmetrical triangle properties, with side ratios that follow the pattern of 1 : √3 : 2.
For the sine function:

• Sine of 30 degrees
  The sine value is \( \frac{1}{2} \)
• Sine of 60 degrees
  The sine value is \( \frac{\sqrt{3}}{2} \)

In the case of the exercise, the sine value of \(-\frac{1}{2}\) is found by reflecting the angle positions into the third and fourth quadrants, showing the importance of special angles in calculation.

The Sine Function in Trigonometry

The sine function is one of the fundamental functions in trigonometry. It traces the y-coordinate of a point on the unit circle as the angle increases.
The function is periodic, with a cycle of 360 degrees (or \(2\pi\) radians). It repeats its values within this range over and over again.
Key features include:

• The maximum value of 1 at 90 degrees (or \(\frac{\pi}{2}\) radians)
• The minimum value of -1 at 270 degrees (or \(\frac{3\pi}{2}\) radians)
• Zero crossings at 0, 180, and 360 degrees (or 0, \(\pi\), and \(2\pi\) radians)

The sine function yields negative values in the third and fourth quadrants, making the understanding of these properties crucial when reflecting angles over quadrants, as seen in the exercise where we adjust angles with negative sines.

Working with Negative Angles

Negative angles represent rotations that occur in the clockwise direction on the unit circle.
They provide a useful way to express the same angle seen while rotating counterclockwise, especially when considering periodic properties of trigonometric functions.
For example, an angle of \(-30^{\circ}\) is the equivalent of a 330-degree counterclockwise rotation, as a full circle is 360 degrees. This equivalency helps in understanding angles across different quadrants, especially when working with periodic trigonometric functions like sine that repeat every 360 degrees.
Converting positive angles to their negative counterparts involves simple subtraction or addition of 360 degrees, facilitating

• the ability to seamlessly switch between representations
• more comprehensive solutions for exercises involving trigonometric identities

Understanding and manipulating negative angles thus extends the versatility of angle calculation in trigonometry.