Why -30 Degrees Is Not A Solution For Sin Θ = -1/2 In [0, 360)
When tackling trigonometric equations, it's crucial to understand the domain in which we're seeking solutions. In this case, we're exploring the equation sin θ = -1/2 within the interval [0°, 360°). This means we're only interested in angles between 0 degrees (inclusive) and 360 degrees (exclusive). The question at hand is: Why isn't -30° a correct answer, even though sin(-30°) does indeed equal -1/2? To understand this, we need to delve into the unit circle, reference angles, and the concept of coterminal angles.
Understanding the Unit Circle and Sine Values
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. The sine of an angle, denoted as sin(θ), corresponds to the y-coordinate of the point where the terminal side of the angle intersects the unit circle. In our specific scenario, where sin θ = -1/2, we're looking for angles whose terminal side intersects the unit circle at a point with a y-coordinate of -1/2. The sine function is negative in the third and fourth quadrants of the unit circle.
To find the angles that satisfy sin θ = -1/2, we first identify the reference angle. The reference angle is the acute angle formed between the terminal side of our angle and the x-axis. We know that sin(30°) = 1/2. Therefore, 30° serves as our reference angle. However, -30° lies in the fourth quadrant, and while it does have a sine of -1/2, it falls outside our desired interval of [0°, 360°). So, while -30° is a valid solution in a broader context, it's not a solution within the specified domain.
Coterminal Angles and the Interval [0°, 360°)
Coterminal angles are angles that share the same terminal side. They differ by multiples of 360°. To find coterminal angles, we can add or subtract 360° (or multiples thereof) from a given angle. While -30° gives us the correct sine value, it's not within our interval. To find coterminal angles within the interval [0°, 360°), we can add 360° to -30°: -30° + 360° = 330°. This gives us one solution within the interval. The angle 330° lies in the fourth quadrant and is a valid solution because it falls within the [0°, 360°) range.
To find the other solution, we consider the third quadrant where the sine function is also negative. We use our reference angle of 30° and add it to 180° (the angle of the negative x-axis) to find the angle in the third quadrant: 180° + 30° = 210°. Thus, 210° is another solution within the interval. This is because 210° corresponds to a point on the unit circle with a y-coordinate of -1/2. Therefore, the two solutions to sin θ = -1/2 within the interval [0°, 360°) are 210° and 330°. In summary, -30° is not a correct answer because it falls outside the specified interval of [0°, 360°). We must find coterminal angles within this interval to obtain the correct solutions.
Why -30° Is Incorrect: A Detailed Explanation
The core reason why -30° is not a correct answer when solving sin θ = -1/2 over the interval [0°, 360°) lies in the defined scope of our solution. We are explicitly seeking solutions that fall within the range of 0 degrees to 360 degrees (excluding 360 degrees itself). This range represents a full rotation around the unit circle, starting from the positive x-axis and moving counterclockwise. Negative angles, like -30°, represent a rotation in the clockwise direction. While they can lead to the same sine value, they are not considered solutions within our specific domain.
To illustrate this further, let's consider the unit circle again. The angle -30° starts at the positive x-axis and rotates 30 degrees in the clockwise direction, landing in the fourth quadrant. This point on the unit circle indeed has a y-coordinate of -1/2, which confirms that sin(-30°) = -1/2. However, when we're restricted to the interval [0°, 360°), we must express this angle as its positive coterminal equivalent. To find this, we add 360° to -30°, resulting in 330°. The angle 330° represents the same terminal side as -30° but is measured in the counterclockwise direction from the positive x-axis, placing it within our desired range.
The importance of the interval restriction is paramount in trigonometric equations. Without it, there would be infinitely many solutions, as we could keep adding or subtracting multiples of 360° and still arrive at angles with the same sine value. The interval [0°, 360°) provides a standard and manageable set of solutions, representing one full rotation around the unit circle. So, while -30° is a valid angle in a general sense, it doesn't meet the specific criteria set by our interval restriction. This highlights the critical role of understanding domain restrictions in solving trigonometric equations accurately.
Finding Correct Solutions within [0°, 360°)
To reiterate, the correct solutions for sin θ = -1/2 within the interval [0°, 360°) are 210° and 330°. These angles satisfy both the equation and the domain restriction. The key to finding these solutions lies in understanding reference angles and the quadrants where the sine function is negative. We established that the reference angle is 30° (since sin(30°) = 1/2). We also know that sine is negative in the third and fourth quadrants. In the third quadrant, we find the solution by adding the reference angle to 180°: 180° + 30° = 210°. In the fourth quadrant, we subtract the reference angle from 360°: 360° - 30° = 330°.
Therefore, -30° is not a valid solution because it's not within the range of [0°, 360°), which is the specified interval for our solutions.
In conclusion, while -30° satisfies the equation sin θ = -1/2, it is not a correct answer when solving over the interval [0°, 360°) because it falls outside this specified range. The correct approach involves identifying the reference angle, determining the quadrants where the sine function has the desired sign, and finding coterminal angles within the given interval. This process ensures that we obtain solutions that are both mathematically accurate and aligned with the defined domain. Understanding these concepts is crucial for successfully navigating trigonometric equations and their applications in various fields of mathematics and science.
