Problem 13 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The exact value of \( \tan^{-1}(\frac{\sqrt{3}}{3}) \) is \( 15^{\circ} \)

Step 1: Identify the input of the inverse function

The input of inverse tangent function \(\tan^{-1}\), in this case, is \(\frac{\sqrt{3}}{3}\).

Step 2: Simplify the input

The \(\sqrt{3}\) can be written as \(\sqrt{3}=\sqrt{2^2-1^2}=\sqrt{4-1}=\sqrt{3}\). Hence, the input of the inverse function is \(\frac{\sqrt{3}}{3} = \frac{\sqrt{3}}{3}\times\frac{2}{2}=\frac{2\sqrt{3}}{6}=\frac{\sqrt{3}}{3}\times 2\), which simplifies into \(\frac{2}{2\sqrt{3}}\).

Step 3: Finding the angle

The procedure to find the corresponding angle involves finding a critical angle in the unit circle for which the tangent equals \(\frac{2}{2\sqrt{3}}\). Remembering that \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), it can be observed that the input is half of this, which suggests that the corresponding angle should be half as well. Hence the angle is \(15^{\circ}\)

Key Concepts

Tangent Function

The tangent function, often abbreviated as tan, is one of the fundamental trigonometric functions. It relates the angle within a right triangle to the ratio of the opposite side over the adjacent side. For any given angle \( \theta \), the tangent can be mathematically expressed as:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]Tangent has interesting properties when extended from a mere right triangle context into any angle measurement using the unit circle.
It is periodic and repeats its values every \( 180^\circ \) or \( \pi \) radians. Additionally, unlike sine and cosine that are defined for all real numbers, the tangent function has undefined values, where its denominator becomes zero (adjacent side equals zero).
These points occur at \( \frac{\pi}{2} + n\pi \), where \( n \) is an integer, making it crucial to manage these undefined areas in calculations, especially for inverse operations.

Unit Circle

A unit circle is a circle with a radius of one unit centered at the origin of a coordinate plane. The unit circle is a significant concept because it allows us to extend trigonometry to work with any angle. Every point on the unit circle corresponds to an angle formed by the positive x-axis and a line connecting the origin to the circle.
On the unit circle:

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.
• Tangent is derived from sine and cosine as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

The unit circle fully wraps around, providing coordinates that allow angles beyond \( 0^\circ \) to \( 360^\circ \) (or \( 0 \) to \( 2\pi \) radians) to be covered with ease, which is critical for working with inverse trigonometric functions like arctan. For instance, the 30-degree angle on the unit circle yields a tangent of \( \frac{1}{\sqrt{3}} \), supporting the solution by finding equivalent angles like 15 degrees.

Angle Measurement

In trigonometry, angles can be measured in degrees or radians, and understanding both forms is fundamental. Degrees, which are more intuitive, divide a circle into 360 parts, while radians focus on the unit circle. A full circle in radians is \( 2\pi \), thus \( 180^\circ \) equals \( \pi \) radians.When utilizing inverse trigonometric functions such as \( \tan^{-1} \), it's important to convert the outputs between radians and degrees for wider applicability:

• \( 30^\circ \) converts to \( \frac{\pi}{6} \) radians.
• \( 15^\circ \) converts to \( \frac{\pi}{12} \) radians.

Recognizing these conversions allows for precise identifications of angles naturally occurring within the unit circle or by inverse function results. In problems like the one given, knowing that \( \tan^{-1} \left( \frac{\sqrt{3}}{3} \right) \) maps to a measured angle ensures your understanding bridges both numeric systems effortlessly.