Problem 46 - A Graphical Approach to Precalculus with Limits

Question

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\tan \frac{x}{2}, \text { given } \tan x=-\frac{\sqrt{5}}{2} \text { and } \frac{\pi}{2} < x < \pi$$

Step-by-Step Solution

Answer

The exact value of \( \tan \frac{x}{2} \) is \( \sqrt{5} \).

Step 1: Understand the Problem

We need to find the value of \( \tan \frac{x}{2} \) given that \( \tan x = -\frac{\sqrt{5}}{2} \) and \( \frac{\pi}{2} < x < \pi \). This means \( x \) is in the second quadrant.

Step 2: Use Identity for \( \tan \frac{x}{2} \)

The half-angle identity for tangent states: \[ \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \text{ or } \frac{\sin x}{1 + \cos x}. \] We will use \( \tan x = \frac{\sin x}{\cos x} \) to find \( \sin x \) and \( \cos x \).

Step 3: Find \( \sin x \) and \( \cos x \) Using \( \tan x \)

We know \( \tan x = -\frac{\sqrt{5}}{2} = \frac{\sin x}{\cos x} \). From the identity \( \sin^2 x + \cos^2 x = 1 \), let's express it in terms of \( \sin x \) and \( \cos x \). Let \( \cos x = a \), then \( \sin x = -\frac{\sqrt{5}}{2} a \) because the tangent is negative in the second quadrant.

Step 4: Solve for \( \cos x \) and \( \sin x \)

Substitute \( \sin x = -\frac{\sqrt{5}}{2}a \) into \( \sin^2 x + \cos^2 x = 1 \): \[ \left( -\frac{\sqrt{5}}{2}a \right)^2 + a^2 = 1. \]This simplifies to \[ \frac{5}{4}a^2 + a^2 = 1. \]Combine terms: \[ \frac{9}{4}a^2 = 1. \]Solve for \(a\): \[ a^2 = \frac{4}{9}, \quad a = \frac{2}{3} \text{ (since \( \cos x \) is negative in the second quadrant)}.\] So, \( \cos x = -\frac{2}{3} \) and \( \sin x = \frac{\sqrt{5}}{3} \).

Step 5: Use Half-Angle Identity

Now we use \( \tan \frac{x}{2} = \frac{\sin x}{1 + \cos x} \) with \( \sin x = \frac{\sqrt{5}}{3} \) and \( \cos x = -\frac{2}{3} \): \[ \tan \frac{x}{2} = \frac{ \frac{\sqrt{5}}{3}}{1 + \left(-\frac{2}{3}\right)} = \frac{\frac{\sqrt{5}}{3}}{\frac{1}{3}} = \sqrt{5}. \]

Step 6: Conclusion

The exact value of \( \tan \frac{x}{2} \) is \( \sqrt{5} \).

Key Concepts

Half-Angle Identity

The Half-Angle Identity is one of the essential tools in trigonometry used to simplify expressions and solve equations involving trigonometric functions. Specifically, it is employed when dealing with angles that are half of a known angle. For the tangent function, the half-angle identity is expressed in two forms:

• \( \tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} \)
• \( \tan \frac{x}{2} = \frac{\sin x}{1 + \cos x} \)

Choosing the appropriate form depends on the context of the problem and the given information.
In our exercise, we had values for \( \sin x \) and \( \cos x \), which makes the second form of the identity quite useful. The Half-Angle Identity lets us shift from a problem involving an angle \( x \) to one involving \( \frac{x}{2} \), effectively allowing calculations where direct measurements could be complex.
The importance of the half-angle identity extends beyond just problem-solving; it also provides insight into the symmetry and relationships within trigonometric functions, highlighting how smaller and covered angles relate to their full counterparts.

Tangent Function

The tangent function, \( \tan x \), is among the six foundational trigonometric functions. It is defined as the ratio of the sine function to the cosine function:

• \( \tan x = \frac{\sin x}{\cos x} \)

This definition is pivotal because it links the behaviors of the sine and cosine functions directly. The tangent function is periodic with a period of \( \pi \), meaning it repeats its values every \( \pi \) radians. This periodicity is critical in simplifying problems involving angles.
Key to the tangent function is understanding its behavior in different quadrants. For instance, in the original exercise, we determined \( \tan x = -\frac{\sqrt{5}}{2} \). This negative value tells us that angle \( x \) lies in a quadrant where sine and cosine have opposite signs—specifically, the second or fourth quadrants.
Solving for \( \tan \frac{x}{2} \) involved finding expressions for \( \sin x \) and \( \cos x \) first. Once determined, these values allowed us to confidently apply the half-angle identity and evaluate the expression, ensuring accurate results.

Second Quadrant

Understanding which quadrant an angle falls into is crucial in trigonometry. Each quadrant has specific characteristics based on the signs of sine and cosine. The second quadrant, where \( \frac{\pi}{2} < x < \pi \), is important for several reasons:

• Sine is positive.
• Cosine is negative.
• The tangent, as \( \frac{\sin x}{\cos x} \), is negative.

Recognizing these characteristics helps in solving exercises accurately, as it guides the selection of the correct forms of identities and ensures that the signs of results are aligned with the quadrant's properties.
In our problem, knowing \( x \) is in the second quadrant was pivotal because it influenced the sign of \( \cos x \), which was used as \( -\frac{2}{3} \) during calculations. This knowledge assured that our solution's direction was correct, permitting the determination of \( \tan \frac{x}{2} \) effortlessly.
Grasping quadrant behaviors equips students with the tools necessary to assess and depict angles correctly, making solutions intuitive and fostering a deeper comprehension of fundamental trigonometric principles.