Problem 90 - A Graphical Approach to Precalculus with Limits

Question

Maximum Viewing Angle The bottom of a 10 -foot-high movie screen is located 2 feet above the eyes of the view ers, all of whom are sitting at the same level. A viewer seated 5 feet from the screen has the maximum viewing angle \(x\) determined by the equation $$\frac{\tan x+0.4}{1-0.4 \tan x}=2.4$$ Find the maximum viewing angle (to the nearest degree).

Step-by-Step Solution

Answer

The maximum viewing angle is approximately 46 degrees.

Step 1: Understand the Equation Given

The equation \( \frac{\tan x + 0.4}{1 - 0.4\tan x} = 2.4 \) is a trigonometric form that can be rewritten using the tangent sum formula. The goal is to solve this equation to find the angle \( x \).

Step 2: Recognize the Form of Tangent Addition

Rearrange the equation to recognize the tangent addition form:\[\tan (x + \theta) = \frac{\tan x + \tan \theta}{1 - \tan x \cdot \tan \theta}\]Here, by comparison, \( \tan \theta = 0.4 \) which implies \( \theta = \tan^{-1}(0.4) \).

Step 3: Solve for x Using Inverse Tangent

Re-express the equation using the angle sum identity:\[\tan(x + \theta) = 2.4\]Solve for \( x \) using inverse tangent:\[x + \theta = \tan^{-1}(2.4)\]Calculate \( \theta \):\[\theta = \tan^{-1}(0.4) \approx 21.8\text{ degrees} \]

Step 4: Isolate and Calculate x

Isolate \( x \) by subtracting \( \theta \) from both sides:\[x = \tan^{-1}(2.4) - \theta\]Calculate: \( \tan^{-1}(2.4) \approx 67.38\text{ degrees} \)So, \(x = 67.38 - 21.8 = 45.58\text{ degrees}\)

Step 5: Round to the Nearest Degree

Round the calculated value of \( x \) to the nearest degree:\[x \approx 46\text{ degrees}\]

Key Concepts

Tangent Addition

In the context of trigonometry, the tangent addition formula helps us identify how to express the tangent of the sum of two angles. Normally, for two angles \(x\) and \(\theta\), this formula is written as:

• \( \tan(x + \theta) = \frac{\tan x + \tan \theta}{1 - \tan x \cdot \tan \theta} \)

This formula proves particularly useful when simplifying complex trigonometric equations. By recognizing the structure of our given equation, we can rewrite the problem using this tangent addition identity.
In our problem, we need to match the terms to identify \(\theta\). Here, it becomes clear from the equation \( \frac{\tan x + 0.4}{1 - 0.4 \tan x} = 2.4 \) that \(\tan \theta = 0.4\).
Using the tangent addition formula allows the equation to be simplified and expressed as \( \tan(x + \theta) = 2.4\), which makes it easier to solve for \(x\).

Inverse Tangent

The inverse tangent, also known as \(\tan^{-1}\) or arctangent, gives us the angle whose tangent is a specific value.

• For example, \(\tan^{-1} (0.4)\) gives the angle whose tangent is 0.4.
• In our exercise, it approximates to around 21.8 degrees.

Using the inverse tangent is crucial for solving trigonometric equations. Given \( \tan(x + \theta) = 2.4 \), it means the angle \(x + \theta\) results in a tangent of 2.4. We use the inverse tangent function to find this specific angle, which is \(\tan^{-1}(2.4)\).
Thus, by calculating both \(\tan^{-1}(2.4)\) and \(\tan^{-1}(0.4)\), we can successfully isolate and solve for \(x\).

Trigonometric Equations

Trigonometric equations involve functions like sine, cosine, and tangent. Solving them requires understanding these functions and their properties. They often serve to find specific angles satisfying certain conditions, such as maximizing the viewing angle in our example.
The original equation \( \frac{\tan x + 0.4}{1 - 0.4\tan x} = 2.4 \) is initially complex, but it can be converted into a simpler form using identities such as tangent addition.

• By applying \( \tan(x + \theta) = 2.4 \), we reduce the problem to simpler terms.
• The final step of isolating \(x\) involves basic algebraic manipulation, i.e., subtracting \(\theta\) from both sides.

These steps simplify the problem-solving process and give the desired angle \(x\), resulting in the maximum viewing angle. Understanding how to manipulate trigonometric equations helps in effectively tackling various real-world applications.