Chapter 3: Problem 55
\(\tan \left[\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right]=\) is (a) \(\frac{6}{17}\) (b) \(\frac{17}{6}\) (c) \(\frac{16}{7}\) (d) \(\frac{7}{16}\)
Short Answer
Expert verified
Question: Find the value of \(\tan \left[\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right]\).
Answer: \(\frac{7}{3}\)
Step by step solution
01
Find \(\tan\left(\cos^{-1}\left(\frac{4}{5}\right)\right)\)
To find this value, let's consider a right triangle with an angle \(\alpha\), such that \(\cos\alpha = \frac{4}{5}\). To find the value of \(\tan\alpha\), we will use the Pythagorean relationship between cosine and sine, given by \(\sin^2\alpha + \cos^2\alpha = 1\), and then find the value of \(\tan\alpha = \frac{\sin\alpha}{\cos\alpha}\).
Let \(\alpha = \cos^{-1}\left(\frac{4}{5}\right)\). Then,
\(\cos\alpha = \frac{4}{5} \implies \cos^2\alpha = \frac{16}{25}\)
Using the Pythagorean relationship, we have:
\(\sin^2\alpha = 1 - \cos^2\alpha = 1 - \frac{16}{25} = \frac{9}{25} \implies \sin\alpha = \frac{3}{5}\)
Now we can find \(\tan\alpha\):
\(\tan\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4}\)
So, \(\tan\left(\cos^{-1}\left(\frac{4}{5}\right)\right) = \frac{3}{4}\).
02
Find \(\tan\left(\tan^{-1}\left(\frac{2}{3}\right)\right)\)
Since \(\tan\left(\tan^{-1}\left(\frac{2}{3}\right)\right)\) is an angle whose tangent is \(\frac{2}{3}\), the value of this expression is simply \(\frac{2}{3}\).
03
Apply the addition formula for tangent
We have \(\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\), where \(\alpha = \cos^{-1}\left(\frac{4}{5}\right)\) and \(\beta = \tan^{-1}\left(\frac{2}{3}\right)\). We have already found the values of \(\tan\alpha\) and \(\tan\beta\).
Let's plug these values into the addition formula for tangent:
\(\tan(\alpha+\beta) = \frac{\frac{3}{4}+\frac{2}{3}}{1-\frac{3}{4}\times\frac{2}{3}} = \frac{\frac{14}{12}}{1-\frac{6}{12}} = \frac{\frac{14}{12}}{\frac{6}{12}} = \frac{14}{6} = \frac{7}{3}\)
The value of \(\tan\left[\cos ^{-1}\left(\frac{4}{5}\right)+\tan ^{-1}\left(\frac{2}{3}\right)\right]=\frac{7}{3}\), which is not listed among the given answer choices. There must be a mistake in the problem statement or answer choices.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions which hold true for any angle. They are essential tools in trigonometry because they help simplify complex trigonometric expressions and solve equations. One fundamental identity is the Pythagorean identity, which states that for any angle \( \theta \), the equation \( \sin^2\theta + \cos^2\theta = 1 \) holds. This identity is derived from the Pythagorean Theorem using a right triangle approach.
By understanding these identities, we can transform and simplify expressions, which is crucial when dealing with inverse trigonometric functions. For instance, identifying the basic relationships helps in deriving values such as \( \sin\theta \), when given \( \cos\theta \), as seen in the solution. Utilizing these identities as building blocks makes it easier to navigate through calculations and reach the desired results.
By understanding these identities, we can transform and simplify expressions, which is crucial when dealing with inverse trigonometric functions. For instance, identifying the basic relationships helps in deriving values such as \( \sin\theta \), when given \( \cos\theta \), as seen in the solution. Utilizing these identities as building blocks makes it easier to navigate through calculations and reach the desired results.
- Pythagorean Identities
- Sum and Difference Formulas
- Double Angle and Half Angle Formulas
Tangent Addition Formula
The tangent addition formula is a specific trigonometric identity that allows you to find the tangent of the sum of two angles. It is given by \( \tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} \). This formula is particularly useful when you need to simplify or find exact values of tangent expressions that are not in their simplest form.
Understanding this formula is crucial, as demonstrated in the exercise solution. By breaking down complex expressions into sums of simpler angles, we can calculate their tan values using known ones. For instance, if you know \( \tan\alpha \) and \( \tan\beta \), this formula helps find \( \tan(\alpha + \beta) \) with ease.
Utilizing the tangent addition formula involves:
Understanding this formula is crucial, as demonstrated in the exercise solution. By breaking down complex expressions into sums of simpler angles, we can calculate their tan values using known ones. For instance, if you know \( \tan\alpha \) and \( \tan\beta \), this formula helps find \( \tan(\alpha + \beta) \) with ease.
Utilizing the tangent addition formula involves:
- Identifying the angles involved
- Calculating the tangent of each angle individually
- Substituting these into the formula
- Simplifying the resulting expression for the final value
Pythagorean Identity
The Pythagorean identity is a vital concept in trigonometry that connects the squares of the sine and cosine of an angle to a constant sum of one. It is expressed as \( \sin^2\theta + \cos^2\theta = 1 \). This identity arises from the sides of a right triangle where the hypotenuse is 1, applying the Pythagorean Theorem within the unit circle.
This identity is not just theoretical but also highly practical. For instance, in our solution process, upon knowing \( \cos\alpha = \frac{4}{5} \), the identity allows us to find \( \sin\alpha \). We calculate \( \sin^2\alpha \) using \( \sin^2\alpha = 1 - \cos^2\alpha \), thus deducing the sine value and thereby solving the problem for \( \tan\alpha \).
Key points of the Pythagorean identity:
This identity is not just theoretical but also highly practical. For instance, in our solution process, upon knowing \( \cos\alpha = \frac{4}{5} \), the identity allows us to find \( \sin\alpha \). We calculate \( \sin^2\alpha \) using \( \sin^2\alpha = 1 - \cos^2\alpha \), thus deducing the sine value and thereby solving the problem for \( \tan\alpha \).
Key points of the Pythagorean identity:
- Allows conversion between sine and cosine.
- Essential for simplifying trigonometric expressions.
- Forms the basis for many other trigonometric identities.