Problem 63 - Precalculus Essentials

Question

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$\tan \left(\cos ^{-1} x\right)$$

Step-by-Step Solution

Answer

The algebraic expression for the equation \(\tan(\cos^{-1}(x))\) is \(\frac{\sqrt{1 - x^2}}{x}\).

Step 1: Establish the relationship between cos and its inverse

Let \(x = \cos(\theta)\), for some angle \(\theta\). Given that this angle lies between 0 and \(\pi\) (because we assume \(x\) is positive), the value of cos at this angle is \(x\) by definition of the inverse function cos. This sets up a right triangle with cosine equal to adjacent/hypotenuse.

Step 2: Find the other side of the triangle

Since we have values for the adjacent side and hypotenuse, we can use the Pythagorean theorem to find the opposite side. The Pythagorean theorem states \[a^2+b^2=c^2\] where \(a\) and \(b\) are the sides of a right triangle and \(c\) is the hypotenuse. Substituting our given values into the theorem gives us \(1-x^2\). In this triangle, the opposite side is therefore \(\sqrt{1 - x^2}\).

Step 3: Use the right triangle to find an algebraic representation

The equation \(\tan(\cos^{-1}(x))\) can be interpreted as finding the tangent of the angle whose cosine is \(x\). By definition, \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\), in our right triangle the opposite is \(\sqrt{1 - x^2}\) and the adjacent is \(x\). Substituting these into the definition gives us \(\frac{\sqrt{1 - x^2}}{x}\). This is the algebraic expression.

Key Concepts

Understanding Right Triangle Relationships

Right triangles, distinguished by their 90-degree angle, are integral to trigonometry. In these triangles, the sides have special names: the longest side opposite to the right angle is called the hypotenuse, while the other two sides are called the legs, with one being adjacent to a given angle and the other opposite to it.

These relationships become useful when using trigonometric functions to determine angles and side lengths. In our exercise, we focus on the cosine of an angle, which is the ratio of the adjacent side to the hypotenuse in a right triangle. Understanding this is key to solving problems involving inverse trigonometric functions and algebraic expressions.

Applying the Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, especially when working with right triangles. It relates the lengths of the sides of a right triangle through the simple equation \[a^2 + b^2 = c^2\], where \(a\) and \(b\) denote the legs, and \(c\) represents the hypotenuse. In our context, knowing one side and the hypotenuse allows us to find the other side through this theorem, thus setting the stage for evaluating trigonometric expressions algebraically.

Interpreting the Tangent Function

The tangent function, often abbreviated as \(\tan\), is another trigonometric function that plays a pivotal role in connecting angles with side lengths. For a given angle in a right triangle, the tangent is the ratio of the opposite side to the adjacent side, expressed as \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).

When dealing with inverse trigonometric functions, we essentially reverse this process. For example, \(\tan(\cos^{-1}(x))\) is asking us to find the tangent of an angle whose cosine is \(x\). By identifying the corresponding sides in the right triangle formed, we can translate this trigonometric function into an algebraic expression.

Formulating Algebraic Expressions

Algebraic expressions are representations of numbers and variables combined using mathematical operations. They form the language through which we can state general mathematical principles and solve various problems, such as those involving inverse trigonometric functions. In our exercise, after establishing the triangle’s sides and applying trigonometric definitions, we arrive at an algebraic expression that simplifies the given trigonometric function, making it easier to understand and calculate with.