Problem 118 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although \(\sin ^{-1}\left(-\frac{1}{2}\right)\) is negative, \(\cos ^{-1}\left(-\frac{1}{2}\right)\) is positive.

Step-by-Step Solution

Answer

The statement makes sense because of the properties and defined ranges of the inverse sine and cosine functions.

Step 1: Understanding Sin^-1

Let's understand \(\sin^{-1}(-\frac{1}{2})\). Inverse sine function or arcsine function - is denoted by \(\sin^{-1}\) or \(\text{asin}\). The principle value of the inverse sine function lies in the interval \([-90°,90°]\) or \([-π/2, π/2]\). So, if the input to \(\sin^{-1}(-\frac{1}{2})\), that is -1/2, lies in the fourth quadrant (where sine is negative), the output will be negative.

Step 2: Understanding Cos^-1

Next, let's understand \(\cos^{-1}(-\frac{1}{2})\). Inverse cosine function or arccosine function - is denoted by \(\cos^{-1}\) or \(\text{acos}\). The range of the inverse cosine function is \([0°,180°]\) or \([0, π]\), and its domain is -1 ≤ x ≤ 1. So, if the input to \(\cos^{-1}(-\frac{1}{2})\), that is -1/2, lies in the second quadrant (where cosine is negative), the output will be positive.

Step 3: Analyzing the statement

The statement says that although \(\sin ^{-1}\left(-\frac{1}{2}\right)\) is negative, \(\cos^{-1}\left(-\frac{1}{2}\right)\) is positive. Considering that \(-\frac{1}{2}\) when input into inverse sine yields a negative result (because \( -\frac{1}{2} \) is in the range of sine in the 4th quadrant which yields negative results) and \(-\frac{1}{2}\) when input into inverse cosine yields a positive result (as it lies in the second quadrant where cosine is negative, but the output of inverse cosine for those values are positive), the statement makes sense and is correct.

Key Concepts

Arcsine Function

The arcsine function, also known as the inverse sine function, is a critical concept in trigonometry. It helps us to determine an angle when the sine value is given. Denoted as \(\sin^{-1}(x)\) or \(\text{asin}(x)\), it answers the question: what angle has this sine value?

For instance, if we have a sine value of \( -\frac{1}{2} \), using the arcsine function, we can find the corresponding angle that produces this sine value. The arcsine function is defined within the interval \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), meaning its values can range from \( -90^\circ \) to \( 90^\circ \). This is why \(\sin^{-1}(-\frac{1}{2})\) results in a negative angle, it lies within the fourth quadrant, where the sine values are negative.

Understanding this function is key to solving various problems involving right-angled triangles and circular functions.

Arccosine Function

Similarly, the arccosine function, or inverse cosine function, notated as \(\cos^{-1}(x)\) or \(\text{acos}(x)\), allows us to find the angle whose cosine value is a given number. When we compute \(\cos^{-1}(-\frac{1}{2})\), we're looking for an angle in the range of \( [0, \pi] \), or \( [0^\circ, 180^\circ] \), where the cosine value is \( -\frac{1}{2} \).

Unlike the arcsine function, the output of the arccosine function can never be negative because it is always an angle in the first or second quadrant, where cosine values can be positive or negative, but angles are always measured positively in the counterclockwise direction. That's why the statement \(\cos^{-1}(-\frac{1}{2})\) brings a positive value, indicating an angle found in the second quadrant.

Trigonometry Principles

Trigonometry is the study of relationships between angles and sides of triangles, particularly right-angled triangles. One of its fundamental principles is that the sine, cosine, and tangent functions relate the angles of a triangle to the lengths of its sides.

The principle values of these functions, defined as inverse trigonometric functions, allow us to reverse this process. Instead of finding the ratio of sides given an angle (as with the sine or cosine functions), we find the angle given a ratio of sides (using arcsine, arccosine, or arctangent functions). It is important to understand that these principles hinge on the ranges and the quadrants in which these functions are valid. This ensures consistent and clear solutions to trigonometric problems.

Quadrants in Trigonometry

In the coordinate system, the quadrants are essential for understanding the signs and values of trigonometric functions. There are four quadrants in trigonometry, each representing a range of angles:\

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• First Quadrant: \(0^\circ \text{ to } 90^\circ\) where all trigonometric values are positive.\
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• Second Quadrant: \(90^\circ \text{ to } 180^\circ\) where only sine values are positive.\
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• Third Quadrant: \(180^\circ \text{ to } 270^\circ\) where only tangent values are positive.\
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• Fourth Quadrant: \(270^\circ \text{ to } 360^\circ\) where only cosine values are positive.\
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\ With this understanding, we can analyze the sine and cosine of an angle depending on the quadrant in which it resides. This conceptual framework helps to explain why, for example, \(\sin ^{-1}(-\frac{1}{2})\) gives a negative angle (due to the angle being in the fourth quadrant), while \(\cos^{-1}(-\frac{1}{2})\) gives a positive angle in the second quadrant where the cosine of angles is negative but the angle measurements are positive.