Problem 14 - Precalculus: A Concise Course

Question

Evaluate the expression without using a calculator. $$ \arctan \sqrt{3} $$

Step-by-Step Solution

Answer

The evaluation of the given expression without a calculator is \(60°\).

Step 1: Identify the Special Triangle

One should recognize that \(\sqrt{3}\) is a value that comes from a special triangle. The 30-60-90 triangle is generally associated with \(\sqrt{3}\). In a 30-60-90 triangle, the sides are in the ratio of 1:\(\sqrt{3}\):2. Therefore, the tangent of the 60 degrees (opposite over adjacent i.e., \(\sqrt{3}\) over 1) is the \(\sqrt{3}\).

Step 2: Evaluate the arctangent

Comparison with the special triangle tells us that \(\arctan(\sqrt{3})\) will yield 60 degrees. The arctangent is the angle for which the tangent yields the specified value. Thus, \(\arctan(\sqrt{3}) = 60°\).

Key Concepts

Special Triangles

Special triangles are important in trigonometry because they allow us to calculate angles and sides using specific, well-known ratios. These triangles have predictable side length relationships that are often used to evaluate trigonometric expressions without a calculator.

There are two main types of special triangles:

• 30-60-90 triangles
• 45-45-90 triangles

The 30-60-90 triangle specifically relates to the problem at hand because it has side length ratios of 1, \( \sqrt{3} \), and 2. This characteristic makes it particularly helpful when evaluating expressions involving \( \sqrt{3} \), like the tangent of certain angles.

By understanding these predefined ratios, we can easily solve many trigonometric problems that involve these special triangles.

Arctangent

The arctangent function, often written as \( \arctan \), is the inverse of the tangent function. It helps us find the angle when we are given the tangent ratio.

In evaluating \( \arctan(\sqrt{3}) \), we are looking for the angle whose tangent is \( \sqrt{3} \). This process involves examining the ratios from a familiar triangle and recognizing which angle matches the given tangent value.

In the context of special triangles, knowing that the tangent of 60 degrees results in \( \sqrt{3} \), the arctangent function helps determine that \( \arctan(\sqrt{3}) = 60^\circ \), utilizing knowledge of special triangle ratios.

30-60-90 Triangle

The 30-60-90 triangle is a fundamental concept in trigonometry. It is characterized by a specific angle configuration and side ratios.

The sides of a 30-60-90 triangle are always in the ratio of 1 : \( \sqrt{3} \) : 2, corresponding to the angles of 30 degrees, 60 degrees, and 90 degrees, respectively. These ratios allow us to easily evaluate trigonometric functions without a calculator.

For example, the side opposite the 60-degree angle is \( \sqrt{3} \) times the side opposite the 30-degree angle. This specific ratio is what connects \( \sqrt{3} \) to the tangent of 60 degrees.

• Regularly used in geometry and trigonometry problems, these triangles simplify many calculations where side lengths and angle measures are involved.
• The side lengths provide a straightforward way to find trigonometric values like sin, cos, and tan for these angles.

Understanding the composition and ratios of the 30-60-90 triangle makes solving related exercises much more intuitive.