Problem 59 Suppose $u=\tan ^{-1} 2$ and \... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

3 Edition

Chapter 5: Problem 59

Suppose $u=\tan ^{-1} 2$ and $v=\tan ^{-1} 3$. Show that $\tan (u+v)=-1$

Short Answer

Expert verified

Using the addition formula for tangent, $\tan(u+v)=\frac{\tan(u)+\tan(v)}{1-\tan(u)\tan(v)}$, we substitute the given values $u=\tan^{-1}{2}$ and $v=\tan^{-1}{3}$ to find $\tan(u+v)=\frac{2+3}{1-2\cdot3}=\frac{5}{-5}=-1$. Thus, $\tan(u+v)=-1$.

Step by step solution

Recall the addition formula for tangent

We will use the formula $\tan(u+v)=\frac{\tan(u)+\tan(v)}{1-\tan(u)\tan(v)}$ to find the value of $\tan(u+v)$.

Substitute the given values of u and v

We are given $u=\tan^{-1}{2}$ and $v=\tan^{-1}{3}$. Therefore, $\tan(u)=2$ and $\tan(v)=3$. Now we can substitute these values into the formula: $\tan(u+v)=\frac{\tan(\tan^{-1}{2})+\tan(\tan^{-1}{3})}{1-\tan(\tan^{-1}{2})\tan(\tan^{-1}{3})}$.

Simplify the expression

Now, we simplify the expression by plugging in the values of $\tan(u)$ and $\tan(v)$: \[\tan(u+v)=\frac{2+3}{1-2\cdot3}=\frac{5}{1-6}.\]

Calculate the final value

Finally, we calculate the value of the expression: \[\tan(u+v)=\frac{5}{-5}=-1.\] Therefore, $\tan(u+v)=-1$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Inverse Tangent

The inverse tangent, also known as arctangent, is the function used to find the angle whose tangent is a given number. When we say $ u = \tan^{-1}(2) $, we mean that $ u $ is the angle whose tangent is 2. Similarly, $ v = \tan^{-1}(3) $ means $ v $ is the angle whose tangent is 3.

This function is particularly useful when working backwards from a known ratio. Here's why:

The range of $ \tan^{-1} $ is between $ -\frac{\pi}{2} $ and $ \frac{\pi}{2} $ radians, ensuring that the angle is determined within this specific interval.
It's commonly used in trigonometric problems to revert from a tangent value back to an angle measurement.

By understanding the role of inverse trigonometric functions like the inverse tangent, you can confidently manipulate equations to solve for unknown angles.

Angle Addition Formula for Tangent

The angle addition formula is a critical trigonometric identity used to find the tangent of the sum of two angles. It is expressed as:\[\tan(u+v) = \frac{\tan(u) + \tan(v)}{1 - \tan(u)\tan(v)}\]This formula works by relating the tangents of the individual angles $ u $ and $ v $ to the tangent of their sum, $ u+v $.

Here are some key points about the formula:

It is derived from the sine and cosine angle addition formulas, demonstrating how different trigonometric functions can interrelate.
This formula allows you to break complex problems into smaller, manageable parts by focusing on known values, such as the tangent of each individual angle.

By substituting the known values of $ \tan(u) $ and $ \tan(v) $ into this formula, you can find the tangent of their sum, which is pivotal in solving the original exercise.

Simplifying Expressions

Simplifying expressions in trigonometry involves reducing the complexity of a given equation to make it easier to solve. In our context, we used the equation:\[\tan(u+v) = \frac{2 + 3}{1 - 2 \cdot 3}\]Let's break down this simplification:

First, evaluate the numerator, $ 2 + 3 = 5 $.
Then, assess the denominator, $ 1 - 2 \cdot 3 = 1 - 6 = -5 $.
By dividing the simplified numerator by the simplified denominator, $ \frac{5}{-5} $, we find the result is $-1$.

The simplification process is crucial for resolving the final value, confirming that $ \tan(u+v) = -1 $. Simplifying expressions properly helps ensure accuracy and clarity in mathematical problems, supporting efficient problem-solving.

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91影视

Suppose \(u=\tan ^{-1} 2\) and \(v=\tan ^{-1} 3\). Show that \(\tan (u+v)=-1\)

Short Answer

Step by step solution

Recall the addition formula for tangent

Substitute the given values of u and v

Simplify the expression

Calculate the final value

Key Concepts

Understanding Inverse Tangent

Angle Addition Formula for Tangent

Simplifying Expressions

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