Problem 71 - Precalculus

Question

Find all possible values of \(\theta,\) where \(0^{\circ} \leq \theta \leq 360^{\circ}\) $$\sin \theta=-\frac{1}{2}$$

Step-by-Step Solution

Answer

\( \theta = 210^{\circ} \) and \( \theta = 330^{\circ} \).

Step 1: Identify Property of Sine Function

Recall that the sine function is negative in the third and fourth quadrants of the unit circle. Hence, the solutions for \( \sin \theta = -\frac{1}{2} \) will be in these quadrants.

Step 2: Determine Reference Angle

The reference angle is the acute angle that the line makes with the x-axis. Knowing that \( \sin 30^{\circ} = \frac{1}{2} \), the reference angle for our function is \( 30^{\circ} \).

Step 3: Find Angle in Third Quadrant

In the third quadrant, the angle can be found by adding the reference angle to 180 degrees: \( 180^{\circ} + 30^{\circ} = 210^{\circ} \).

Step 4: Find Angle in Fourth Quadrant

In the fourth quadrant, the angle can be found by subtracting the reference angle from 360 degrees: \( 360^{\circ} - 30^{\circ} = 330^{\circ} \).

Key Concepts

Sine function

The sine function is a fundamental concept in trigonometry and is defined as the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. It is part of the set of trigonometric functions which also includes cosine and tangent. The sine function is usually written as \( \sin \theta \), where \( \theta \) is the angle. In the context of the unit circle, the sine function represents the y-coordinate of a point on the circle.

Some important characteristics of the sine function include:

• Its periodic nature, repeating every \(360^{\circ}\) or \(2\pi\) radians.
• The range of its values lies between -1 and 1, inclusive.
• It equals zero at \(\theta = 0^{\circ}, 180^{\circ}, 360^{\circ},\) etc.

The fact that \( \sin \theta = -\frac{1}{2} \) leads us to search for angle solutions in specific quadrants where sine is negative, particularly the third and fourth quadrants.

Unit circle

The unit circle is a circle with a radius of 1 unit that is centered at the origin (0,0) of a coordinate system. This tool is crucial for understanding trigonometric functions, as it allows us to define these functions for any real number angle.

When using the unit circle to solve problems:

• The angle \( \theta \) is measured from the positive x-axis, moving counter-clockwise.
• The coordinates of any point on the unit circle can be expressed as \((\cos \theta, \sin \theta)\), encapsulating both the cosine and sine functions.
• Different quadrants of the unit circle indicate the sign of the trigonometric functions.

As related to the original exercise, the negative sine value indicates we are looking in the third and fourth quadrants, where sine values are negative. Here, we find our solutions by considering these particular regions.

Reference angle

In trigonometry, a reference angle is the acute angle formed by the terminal side of an angle \( \theta \) and the x-axis. Essentially, it is the "smallest" angle that provides the same sine, cosine, and tangent values as the original angle, but always positive. This concept is particularly useful when solving trigonometric equations.

Important properties of reference angles include:

• The reference angle is always between \(0^{\circ}\) and \(90^{\circ}\).
• The value of a sine function at a given angle is the same as its sine value at its reference angle, if the sign of the sine is considered.
• To find angles in other quadrants, you either add or subtract the reference angle from \(180^{\circ}\) or \(360^{\circ}\), depending on which quadrant you're dealing with.

For our exercise, knowing \( \sin 30^{\circ} = \frac{1}{2} \), the 30-degree reference angle helps us find \(\theta\) when \( \sin \theta = -\frac{1}{2} \) by considering both the third and fourth quadrants, yielding\( \theta = 210^{\circ} \) and \(330^{\circ}\) as possible solutions.