Problem 3 - Algebra 2 and Trigonometry

Question

Is arctan \(1=220^{\circ}\) a true statement? Justify your answer. \(y=\arcsin \frac{1}{2}\)

Step-by-Step Solution

Answer

The statement is false because arctan(1) is 45°.

Step 1: Define arctan

The arctan or inverse tangent function, \ \(y = \arctan(x)\), represents the angle \(y\) whose tangent is \(x\). For a given \(x\), the principal value of \(\arctan(x)\) is restricted to \(-\frac{\pi}{2} < y < \frac{\pi}{2}\) or \(-90^{\circ} < y < 90^{\circ}\).

Step 2: Evaluate arctan(1)

The angle whose tangent is 1 is 45 degrees or \(\frac{\pi}{4}\) radians. Therefore, \(\arctan(1) = 45^{\circ}\) or \(\frac{\pi}{4}\).

Step 3: Compare with 220 degrees

Since the range for \(\arctan(x)\) is restricted to \(-90^{\circ} < y < 90^{\circ}\), it cannot equal \(220^{\circ}\). Therefore, the statement \(\arctan(1) = 220^{\circ}\) is false.

Key Concepts

Understanding the Arctan Function

The arctan function, also known as the inverse tangent function, is a critical concept in trigonometry. It helps us find the angle associated with a given tangent value. We denote it as \( y = \arctan(x) \), which means that \( y \) is the angle whose tangent is \( x \). This function is particularly useful when you have the value of tangent and you want to find out the corresponding angle.

Unlike the tangent function, which can output any real number, the arctan function is specifically designed to produce angles between \(-\frac{\pi}{2} \) and \( \frac{\pi}{2} \) radians, or equivalently, between \(-90^\circ \) and \( 90^\circ \). This range ensures a unique angle output for every possible input value of \( x \). In other words, no matter the input, arctan will always output an angle within this narrow band.

Therefore, if you calculate \( \arctan(1) \), you're essentially looking for an angle \( y \) such that \( \tan(y) = 1 \). The result is \( \frac{\pi}{4} \) radians or \( 45^\circ \). Thus, statements suggesting an arctan value outside of this principal range, such as \( 220^\circ \), are incorrect.

What is Principal Value?

In trigonometry, the concept of the "principal value" refers to the range within which inverse trigonometric functions like arctan are defined. It helps us assign a single, consistent output to the function regardless of the input size. For the arctan function, the principal value is limited to the interval \(-\frac{\pi}{2} < y < \frac{\pi}{2} \) or \(-90^\circ < y < 90^\circ \).

Think of the principal value as a window that provides a "standard" answer for trigonometric problems, allowing you to solve them consistently and avoid ambiguities. Without this specific range, an inverse function could potentially have numerous value outputs, making calculations confusing.

For example, when solving \( \arctan(1) \), many angles \( y \) could satisfy \( \tan(y) = 1 \), such as \( 45^\circ, 225^\circ, 405^\circ \), etc. However, only \( 45^\circ \) falls within the principal value range. That's why \( \arctan(1) = 45^\circ \) is correct, while values like \( 220^\circ \) are not valid, based on the principal value.

The Role of the Tangent Function

The tangent function is fundamental in trigonometry, often abbreviated as \( \tan(\theta) \), where \( \theta \) is an angle. Its role is to calculate the ratio of the opposite side to the adjacent side in a right-angled triangle. In terms of radians, this function is periodic, repeating its values every \( \pi \) radians, or \( 180^\circ \).

One of the key attributes of the tangent function is that it can take any and all real number values, ranging from negative to positive infinity. However, when you move to its inverse, the arctan function, you're looking at a very specific subset of these values (the principal value) that give you a unique angle.

The angle where \( \tan(y) = 1 \) is particularly notable, as it equals \( 45^\circ \) in the principal value interval. This is because \( \tan(45^\circ) = 1 \) due to the equal length of the opposite and adjacent sides in a right-angled isosceles triangle.

When considering the statement \( \arctan(1) = 220^\circ \), it's clear that the principal value range does not support this claim, since \( 220^\circ \) is outside of the eligible domain of effort for arctan. Hence, the assertion is false.