Problem 13 - Precalculus Essentials

Question

Find the exact value of each expression. $$\tan ^{-1} \frac{\sqrt{3}}{3}$$

Step-by-Step Solution

Answer

\(\tan^{-1}\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{6}\) radians or 30 degrees

Step 1: Understanding the exercise

The exercise asks to find the exact value of \( \tan^{-1}(\frac{\sqrt{3}}{3}) \). The tangent of what angle equals \( \frac{\sqrt{3}}{3} \)? This step requires recalling or understanding the specific tangent values of a 30-60-90 triangle.

Step 2: Identifying the corresponding angle

Knowing the properties of the 30-60-90 triangle, the tangent of 30 degrees or \( \frac{\pi}{6} \) radians equals to \( \frac{1}{\sqrt{3}} \), which simplifies to \( \frac{\sqrt{3}}{3} \). Therefore, \( \tan^{-1}(\frac{\sqrt{3}}{3}) \) corresponds to an angle of 30 degrees or \( \frac{\pi}{6} \) radians.

Key Concepts

Understanding the Tangent Function

The tangent function is a fundamental concept in trigonometry that relates the angles of a right-angled triangle to the ratio of its opposite side to the adjacent side. Specifically, for a given angle \( \theta \), the tangent function, denoted as \( \tan(\theta) \), is defined as the ratio of the length of the side opposite to \( \theta \) over the length of the side adjacent to \( \theta \) in a right-angled triangle.

When it comes to the inverse of the tangent function, designated as \( \tan^{-1} \) or \( \arctan \), we are looking for an angle whose tangent is a particular value. For instance, if we have \( \tan\theta = \frac{a}{b} \), then \( \theta = \tan^{-1}(\frac{a}{b}) \).

Real-Life Applications

Knowledge of the tangent function is not just academic; it's utilized in various fields such as architecture, physics, and engineering to calculate slopes, angles, and distances. Understanding how to manipulative and invert the tangent function is essential for solving problems in these areas.

The Geometry of a 30-60-90 Triangle

A 30-60-90 triangle is a special type of right triangle characterized by its angle measurements: one angle measures 30 degrees, another 60 degrees, and the right angle is 90 degrees. The lengths of the sides of a 30-60-90 triangle follow a specific ratio. If the shortest side, opposite the 30-degree angle, is of length x, then the side opposite the 60-degree angle, also known as the hypotenuse, will be 2x, and the remaining side, opposite the 90-degree angle, will be \( x\sqrt{3} \).

These properties make 30-60-90 triangles extremely useful for solving problems without needing calculators, as their angles and side ratios are consistent and easily remembered. They are essential for understanding certain trigonometric ratios and functions without needing to measure every single time.

Practical Use in Trigonometry

Given the recurring nature of angles like 30 and 60 degrees in various shapes and patterns, the 30-60-90 triangle serves as a quick reference for solving more complex geometric and trigonometric problems.

Radians: An Alternative Measure of Angles

Radians are a unit of angular measure used in mathematics, particularly in trigonometry and calculus. Unlike degrees, which are based on dividing a circle into 360 equal parts, radians are based on the radius of a circle. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

The circle is a central figure in trigonometry, and using radians provides a way to directly relate angular measures to the circle's radius, simplifying many types of calculations. There are \( 2\pi \) radians in a full circle, so 180 degrees is equivalent to \( \pi \) radians. This means that commonly used angles in trigonometry such as 30 degrees can be converted to radians; in this case, 30 degrees is the same as \( \frac{\pi}{6} \) radians.

Advantages in Calculation

Using radians can sometimes simplify trigonometric expressions and integrations in calculus. It's a natural unit that connects linear and angular measurements, which is essential when dealing with circular motion or waveform analysis in physics and engineering.