Problem 73 - Precalculus: A Concise Course

Question

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example \(7 .)\) $$ \tan \left(\arccos \frac{x}{3}\right) $$

Step-by-Step Solution

Answer

\The equivalent algebraic expression to \(\tan(\arccos(\frac{x}{3})\)\) is \(\frac{\sqrt{9 - x^2}}{x}\)

Step 1: Draw a right triangle

Draw a right triangle and label the adjacent side as \(x\), the hypotenuse as \(3\) and the angle between them as \(\theta\). Identify that \(\theta = \arccos(\frac{x}{3})\).

Step 2: Determine the length of the opposite side

From the Pythagorean theorem, let \(y\) be the length of the opposite side. Then, we know that \(3^2 = x^2 + y^2\). Therefore, \(y^2 = 9 - x^2 \) which will result in \(y = \sqrt{9 - x^2 }\).

Step 3: Find the equivalent tangent expression

The tangent of the angle \(\theta\) is defined as the ratio between the opposite side and the adjacent side, which results in: \(\tan(\theta) = \frac{y}{x} = \frac{\sqrt{9 - x^2}}{x}\). Since \(\theta\) is equivalent to \(\arccos(\frac{x}{3})\) the expression \(\tan(\arccos(\frac{x}{3}))\) is equivalent to \(\frac{\sqrt{9 - x^2}}{x}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are pivotal in solving various mathematical problems, especially when dealing with right triangles. They serve as the reverse operations of the standard trigonometric functions and are denoted by prefixes like 'arc' or the symbol '^{-1}'. For instance, the inverse of sine, cosine, and tangent functions are written as \(\arcsin\), \(\arccos\), and \(\arctan\) respectively.

These functions allow us to find angles when the ratios of the sides of a right triangle are known. For example, if we know the adjacent side (\(x\)) and the hypotenuse (\(3\)) of a right triangle, we can find the angle \(\theta\) using \(\arccos\) as \(\theta = \arccos\left(\frac{x}{3}\right)\).

Understanding how to use these functions is crucial because they form the basis for solving geometric problems, analyzing periodic phenomena, and even in calculus for finding derivatives and integrals of trigonometric functions.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics, particularly in geometry. It states that, in a right triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)): \[c^2 = a^2 + b^2\].

This theorem serves as an essential tool for finding missing lengths in right-angled triangles, which is a common occurrence in a wide array of mathematical applications. In the given exercise, knowing that the hypotenuse is \(3\) and one of the sides (\(x\)) allows us to apply the theorem to find the length of the opposite side (\(y\)). This relationship ensures that we have all the necessary information to find the ratio of the sides for our trigonometric calculations.

Right Triangle Trigonometry

Right triangle trigonometry is the branch of trigonometry that focuses on right triangles – triangles with one angle measuring 90 degrees. Each trigonometric function relates an angle to the ratios of two sides of a right triangle.

The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). For the tangent, the ratio is the length of the side opposite the angle to the length of the side adjacent to it (\(\tan(\theta) = \frac{opposite}{adjacent}\)).

In the context of the exercise, we're dealing with \(\tan(\theta)\). With \(\theta\) given as \(\arccos(\frac{x}{3})\), we use the concept of inverse trigonometric functions to understand the relationship between the angle and its cosine, and consequently, determine the tangent using the lengths of the triangle's sides determined through the Pythagorean theorem. This interconnectedness of trigonometric identities and the Pythagorean theorem exemplifies the elegance and utility of right triangle trigonometry in solving algebraic expressions involving trigonometric values.