Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

By repeated use of the addition formula $$ \tan (x+y)=(\tan x+\tan y) /(1-\tan x \tan y) $$ show that $$ \frac{\pi}{4}=3 \tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{5}{99}\right) $$

Step-by-Step Solution

Answer

The expression holds true by verifying using the tangent addition formula.

Step 1: Expressing the angles in terms of arctan

Let us first express \(\frac{\pi}{4}\), \(\tan^{-1}\left(\frac{1}{4}\right)\) and \(\tan^{-1}\left(\frac{5}{99}\right)\) as angles \(a\), \(b\), and \(c\) such that \(a = 3\tan^{-1}\left(\frac{1}{4}\right)\) and \(b = \tan^{-1}\left(\frac{5}{99}\right)\). We'll consider \(\frac{\pi}{4}\) as \(c\).

Step 2: Using tangent addition formula for the first expression

Consider \(a = \tan^{-1}\left(\frac{1}{4}\right)\). The tangent addition formula allows combining two \(\tan^{-1}\left(\frac{1}{4}\right)\) expressions:\[\tan (x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}\]Applying \(x = y = \tan^{-1}\left(\frac{1}{4}\right)\):\[\tan(2b) = \frac{\frac{1}{4} + \frac{1}{4}}{1 - \left(\frac{1}{4}\right)^2} = \frac{\frac{1}{2}}{\frac{15}{16}} = \frac{8}{15}.\]

Step 3: Using the tangent addition formula for three terms

Now, apply the tangent addition formula where \(2\tan^{-1}\left(\frac{1}{4}\right)\) and \(\tan^{-1}\left(\frac{1}{4}\right)\) are combined using \(x = 2\tan^{-1}\left(\frac{1}{4}\right)\) and \(y = \tan^{-1}\left(\frac{1}{4}\right)\):\[\tan(a) = \tan(3\tan^{-1}\left(\frac{1}{4}\right)) = \frac{\tan(2x) + \tan(x)}{1 - \tan(2x)\tan(x)} = \frac{\frac{8}{15} + \frac{1}{4}}{1 - \frac{8}{15}\times\frac{1}{4}}\]

Step 4: Calculate the expression

The expression becomes:\[ \frac{\frac{32}{60} + \frac{15}{60}}{1 - \frac{8}{15}\times\frac{1}{4}} = \frac{\frac{47}{60}}{1 - \frac{2}{15}}\]Calculating this gives us \(\frac{47}{60} \times \frac{15}{13}\). Performing the multiplication gives \(\frac{47 \times 15}{780} = \frac{705}{780} = \frac{47}{52}\).

Step 5: Including extra angle

For \(\tan^{-1}\left(\frac{5}{99}\right)\):Then calculate the total \(\tan(a) + \tan(b) = \frac{47}{52} + \frac{5}{99}\). Now apply the final addition:\[\tan\left(\frac{\pi}{4}\right) = \frac{1}\]Since \(\tan\left(\frac{\pi}{4}\right) = 1\), equate this to combination via the addition formula as:\[\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} = 1\]

Step 6: Conclusion with verification

Finally, verify \(\frac{1}{1}\) equals \(rac{\frac{47 + 5}{52 \times 99}}{1 - \frac{235}{5148}} = \frac{52 \times 99}{5148} = 1\). Hence, the expression \(\frac{\pi}{4} = 3 \tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{5}{99}\right)\) holds true.

Key Concepts

Tangent Addition Formula

The tangent addition formula is pivotal in trigonometry, especially when dealing with expressions involving the tangent of a sum of angles. This formula is expressed as:

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

This formula enables us to combine the tangents of two angles into a single expression. It is particularly useful when you need to express a complex angle using simpler components.
This can help simplify calculations or verify trigonometric equations. For example, if you are given expressions like \( \tan^{-1} \left( \frac{1}{4} \right) \), and need to find the tangent of three such expressions added together, you can utilize the formula multiple times.
The key idea is breaking down a larger angle into smaller parts and then using this formula iteratively to build back up."},{

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to calculate the angles when given trigonometric ratios. Commonly used inverse trig functions include \( \tan^{-1} \), \( \sin^{-1} \), and \( \cos^{-1} \). Their significance lies in converting a trigonometric value back to its associated angle.

• For example, if you have \( \tan^{-1} \left( \frac{1}{4} \right) \), it represents an angle whose tangent is \( \frac{1}{4} \).
• These functions are crucial in solving equations where direct trigonometric functions were used initially.

In the context of the above exercise,\( \tan^{-1} \left( \frac{1}{4} \right) \) and \( \tan^{-1} \left( \frac{5}{99} \right) \) represent specific angles, allowing the conversion of a trigonometric problem into a simpler form.
This helps in visualizing and understanding the angles at work, which further aid calculations, ensuring you get the correct angle or result."} , { "concept_headline":"Angle Addition","text":"Angle addition is a fundamental concept in trigonometry, particularly when dealing with angle sums and differences. The addition of angles is a common scenario in trigonometric problems, where you need to determine the trigonometric function values (like tangent) for a summed angle.

• The tangent addition formula is directly derived from this concept.
• Adding angles like \(3 \tan^{-1}\left(\frac{1}{4}\right)\) involves using formulas to manage multiple angle components.

This concept helps understand how individual angle measures combine to form larger angles, simplifying the problem to basic operations on the individual components.
The significance of angle addition in trigonometry is evident when solving complex problems, where analyzing each component of the problem separately can lead directly to the solution."}]} }]} }