Problem 33 - Algebra and Trigonometry

Question

Rewrite the expression as an algebraic expression in x. $$ \cos \left(\sin ^{-1} x\right) $$

Step-by-Step Solution

Answer

\( \cos(\sin^{-1}(x)) = \sqrt{1 - x^2} \)

Step 1: Understand Inverse Trigonometric Functions

The expression involves the inverse sine function, denoted as \( \sin^{-1}(x) \). This function returns an angle \( \theta \) such that \( \sin(\theta) = x \), where \( \theta \) is measured in radians between \(-\frac{\pi}{2}\) and \( \frac{\pi}{2}\).

Step 2: Recognize the Trigonometric Identity

Recall the identity \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \) for any angle \( \theta \). This identity will help in connecting \( \cos(\sin^{-1}(x)) \) to a simple expression in terms of \( x \).

Step 3: Apply the Identity

Since \( \theta = \sin^{-1}(x) \) implies \( \sin(\theta) = x \), we apply the identity:\[ \cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - x^2} \]Therefore, \( \cos(\sin^{-1}(x)) = \sqrt{1 - x^2} \).

Step 4: Write the Final Expression

From the previous step, we obtained the expression \( \cos(\sin^{-1}(x)) = \sqrt{1 - x^2} \). This is the simplified algebraic expression for the original trigonometric expression.

Key Concepts

Trigonometric Identities

Trigonometric identities are key to simplifying complex trigonometric expressions. They are equations that hold true for all possible values of the involved variables. Two of the most important and frequently used identities are the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and the related identity for cosine, \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \).
These identities allow you to express one trigonometric function in terms of another, which is particularly useful when dealing with inverse trigonometric functions as seen in the solution above. The beauty of these identities is that they connect the sine and cosine functions through the concept of angles and the unit circle.
Moreover, understanding and recognizing these identities is crucial for solving a wide array of problems in trigonometry. The identity \( \cos(\sin^{-1}(x)) = \sqrt{1 - x^2} \) transforms the inverse sine function into its equivalent, simpler expression in terms of \( x \). This makes trigonometric expressions not just easier to handle, but also more straightforward to analyze.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations like addition, subtraction, multiplication, and division. When we talk about converting a trigonometric or inverse trigonometric expression into an algebraic expression, we're transforming it into a form that only involves variables and arithmetic operations. This is what was accomplished in the exercise by rewriting \( \cos(\sin^{-1}(x)) \) as \( \sqrt{1 - x^2} \).
To understand this process, keep in mind that \( x \) represents numbers that fall within specific ranges depending on the function. For \( \sin^{-1}(x) \), this means \( -1 \leq x \leq 1 \). The resulting expression, \( \sqrt{1 - x^2} \), is an algebraic expression. It no longer explicitly shows the trigonometric function but maintains the relationship via algebraic terms.
This transformation is immensely helpful because algebraic expressions are often simpler to work with than trigonometric expressions, especially when evaluating at particular values or when used in further calculations, such as integration or differentiation.

Sine and Cosine Functions

Sine and cosine are foundational functions in trigonometry and are often introduced using the unit circle. The sine function gives the y-coordinate, while the cosine function gives the x-coordinate of a point on the unit circle corresponding to a given angle.
In our exercise, we begin with the inverse sine function, \( \sin^{-1}(x) \), which gives us an angle \( \theta \) such that \( \sin(\theta) = x \). This specific angle is restricted to lie between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), ensuring the function is one-to-one and thus invertible.
Using the Pythagorean identity \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \), we transition from sine to cosine. This switch utilizes how these functions relate through an angle’s sine and cosine values. By recognizing this relation, we see that \( \cos(\sin^{-1}(x)) \) simplifies to an expression in terms of \( x \), showing the profound interplay between both trigonometric functions and their inverses.