Problem 6 - Algebra 2 and Trigonometry

Question

Is arctan \(1=220^{\circ}\) a true statement? Justify your answer. \(y=\arctan \sqrt{3}\)

Step-by-Step Solution

Answer

No, \(\arctan 1\) is \(45^{\circ}\), not \(220^{\circ}\).

Step 1: Understand arctan Function

The arctan function, or inverse tangent function, is used to find the angle whose tangent is a given number. For example, if \( y = \arctan(x) \), then \( \tan(y) = x \) and \(-\dfrac{\pi}{2} < y < \dfrac{\pi}{2} \) (in radians).

Step 2: Check if \(\arctan 1 = 220^{\circ}\)

First, recall that the arctan function returns angles in the range \(-90^{\circ}\) to \(90^{\circ}\). The tangent of \(45^{\circ}\) is 1, meaning \(\arctan 1 = 45^{\circ}\). Therefore, \(\arctan 1 = 220^{\circ}\) is incorrect because 220 degrees is outside the range of the arctan function.

Step 3: Evaluate \(y=\arctan \sqrt{3}\)

Since \( y = \arctan \sqrt{3} \), we find that \( \tan^{-1} \sqrt{3} \) corresponds to the angle whose tangent is \(\sqrt{3}\). In a 30-60-90 triangle, the tangent of \(60^{\circ}\) (or \(\dfrac{\pi}{3}\) radians) is \(\sqrt{3}\). Thus, \(y = 60^{\circ}\).

Key Concepts

Arctan Function

The arctan function, known as the inverse tangent, is a fundamental concept in trigonometry. It's used to determine an angle when the tangent value is known.

• Consider a right triangle where the opposite side is measured relative to the adjacent side. The ratio of these two sides gives us the tangent of an angle.
• By applying the arctan function, we find the specific angle whose tangent equals that ratio.

In formal terms, if you have \[ y = \arctan(x) \]then it follows that \[ \tan(y) = x \]. This means you're solving for angle \(y\) that has a tangent of \(x\).It's important to remember that the arctan function provides angles specifically within a restricted range, as it doesn’t return angles beyond its principal value domain.

Angle Measurement

Angle measurement can be expressed in two principal units: degrees and radians. Both units are used interchangeably in trigonometry but understanding their conversion is key.

• Degrees are more common in everyday use. A full circle is 360 degrees.
• Radians, often used in mathematics and physics, express angles as real numbers, with a complete circle being \(2\pi\) radians.

A simple conversion factor is that 180 degrees is equivalent to \(\pi\) radians, making:\[1^{\circ} = \frac{\pi}{180} \, \text{radians}\]When working with inverse trigonometric functions like arctan, the function typically returns angles in radians unless specified otherwise. In the context of \( \arctan(1)\), this corresponds to \(45^{\circ}\) or \(\dfrac{\pi}{4}\) in radians.

Range of Arctan

The range of the arctan function is crucial for interpreting its output correctly. The range determines the set of possible angles the function can return and is from \(-\dfrac{\pi}{2} \, \text{to} \, \dfrac{\pi}{2}\) in radians, equivalent to \(-90^{\circ} \, \text{to} \, 90^{\circ}\).

• This limitation ensures a unique angle for any given tangent value since the tangent function is periodic and repeats every 180 degrees.
• By using this range, arctan avoids ambiguity when identifying angles corresponding to a given tangent.

For example, even if multiple angles could have the same tangent value, \( \arctan(x) \) will always select the angle within its primary range. This means, crucially, that statements like \(\arctan(1)=220^{\circ}\) are incorrect, since 220 degrees falls outside the range of \(-90^{\circ} \, \text{to} \, 90^{\circ}\). Instead, \( \arctan(1) \) correctly returns \(45^{\circ}\).