Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\tan ^{-1} \sqrt{3}$$

Step-by-Step Solution

Answer

The exact value of \( y \) is \( \frac{\pi}{3} \).

Step 1: Understanding the Problem

We need to find the exact value of the angle \( y \) such that \( y = \tan^{-1} \sqrt{3} \). This means we are looking for an angle whose tangent is \( \sqrt{3} \).

Step 2: Recall Trigonometric Values

Recall the basic trigonometric value, \( \tan \theta = \sqrt{3} \), which is a known value from the unit circle or trigonometric ratios. Specifically, \( \tan(\frac{\pi}{3}) = \sqrt{3} \).

Step 3: Determine the Principal Value

The function \( \tan^{-1} \) gives the principal value of the angle between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \). Since \( \frac{\pi}{3} \) is within that range, it is the exact angle we need.

Key Concepts

Trigonometric Ratios

Trigonometric ratios are fundamental in understanding angles and their relationships to circles, triangles, and geometry. The primary trigonometric ratios are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). These ratios are based on the sides of right triangles:

• Sine of an angle is the ratio of the length of the opposite side to the hypotenuse
• Cosine is the ratio of the adjacent side to the hypotenuse
• Tangent is the ratio of the opposite side to the adjacent side

For example, in the exercise, we determined \( \tan(\frac{\pi}{3}) = \sqrt{3} \). This tells us that when the opposite side is \( \sqrt{3} \) times the length of the adjacent side, the angle is \( \frac{\pi}{3} \). Recognizing these standard trigonometric values without calculation is crucial for solving inverse trigonometric functions quickly.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a powerful tool in trigonometry for visualizing angles and their trigonometric function values.

• On the unit circle, the x-coordinate represents \( \cos \theta \)
• The y-coordinate is \( \sin \theta \)
• The ratio \( \frac{\sin \theta}{\cos \theta} \) gives the tangent, \( \tan \theta \)

In the unit circle, common angle measures such as \( \frac{\pi}{3}, \frac{\pi}{4}, \frac{\pi}{6} \), etc., have corresponding trigonometric values. In the exercise, finding \( \tan^{-1}(\sqrt{3}) \) means locating the angle whose tangent equals \( \sqrt{3} \), which happens at \( \frac{\pi}{3} \) on the unit circle. The unit circle helps in identifying these key angles and their respective trigonometric ratios.

Principal Value

The concept of principal value is central to solving inverse trigonometric functions. Inverse trigonometric functions provide the angle whose trigonometric ratio is a given number. However, since trigonometric functions are periodic, an infinite number of angles could produce the same value. Therefore, we define principal values to limit this to a specific range.

• For \( \tan^{-1} x \)
• The principal value lies between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \)
• This range includes angles from the first quadrant, where all trigonometric values are positive

In our specific exercise, \( \frac{\pi}{3} \) falls within this interval, confirming it as the principal value for \( \tan^{-1}(\sqrt{3}) \). Understanding principal values aids in narrowing down to the correct angle when interpreting inverse trigonometric expressions.