Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

find the exact value without using a calculator. $$ \arctan (\sqrt{3}) $$

Step-by-Step Solution

Answer

\( \arctan(\sqrt{3}) = \frac{\pi}{3} \).

Step 1: Understand the Inverse Trigonometric Function

The function \( \arctan(x) \) gives the angle \( \theta \) such that \( \tan(\theta) = x \). We are asked to find the angle whose tangent equals \( \sqrt{3} \).

Step 2: Recall Special Angle Values of Tangent

The tangent function has known values for certain angles. Recall that \( \tan(30^\circ) = \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \), \( \tan(45^\circ) = \tan\left(\frac{\pi}{4}\right) = 1 \), and \( \tan(60^\circ) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \).

Step 3: Find Corresponding Angle for \( \sqrt{3} \)

From our recall in Step 2, \( \tan(60^\circ) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \). Therefore, the angle corresponding to \( \arctan(\sqrt{3}) \) is \( \frac{\pi}{3} \).

Step 4: Verify the Range of \( \arctan \)

The range of the \( \arctan \) function is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), which includes \( \frac{\pi}{3} \). This confirms that \( \frac{\pi}{3} \) is a valid solution.

Key Concepts

Arctangent

The arctangent function, often written as \( \arctan(x) \), is an inverse trigonometric function. It asks the question: "What angle \( \theta \) has a tangent equal to \( x \)?" Simply put, if you know the tangent value and want to find the angle, you'll use \( \arctan \).

The \( \arctan \) function returns an angle in the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), which means it gives angles in the range from negative 90 degrees to positive 90 degrees. This helps ensure we get one specific angle from many possibilities.

For instance, if you have \( \arctan(\sqrt{3}) \), you're looking for the angle where the tangent function equals \( \sqrt{3} \). After exploring the trigonometric special angles, you'll find that it is \( \frac{\pi}{3} \) radians or 60 degrees.

Tangent Function

The tangent function, expressed as \( \tan(\theta) \), relates an angle of a right triangle to the ratio of the opposite side over the adjacent side. This function repeatedly cycles through its values, creating a pattern in what is called its periodic nature.

The tangent function repeats every \( \pi \) radians or 180 degrees, which means its value pattern emerges after these intervals.

• It's essential to understand that \( \tan \theta \) can equal the same value for multiple angles because of this periodic repetition.
• A nice feature of the tangent function is its vertical asymptotes, occurring at angles such as \( \frac{\pi}{2} \), \( \frac{3\pi}{2} \), etc. This reflects how the function grows larger and suddenly switches to negative values.

Knowing these aspects helps when working with equations involving \( \tan \), such as solving for \( \arctan(\sqrt{3}) \) and identifying consistent angle solutions.

Special Angle Values

Trigonometric functions like tangent have specific, easily remembered values at certain common angles. These are often called special angle values, and they appear frequently:

• At \( 30^\circ \) \( \left(\frac{\pi}{6}\right) \), \( \tan \) is \( \frac{1}{\sqrt{3}} \).
• At \( 45^\circ \) \( \left(\frac{\pi}{4}\right) \), \( \tan \) is 1.
• At \( 60^\circ \) \( \left(\frac{\pi}{3}\right) \), \( \tan \) is \( \sqrt{3} \).

Using these special angle values simplifies calculations without a calculator. They serve as a handy reference when deducing angles in inverse functions. Thus, when you see \( \arctan(\sqrt{3}) \), you can determine its equivalent angle as \( \frac{\pi}{3} \), simply because it matches the special angle value of 60 degrees without additional tools.