Problem 49 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

Question: Write the expression \(\cos \left(\sin ^{-1} v\right)\) as an algebraic expression in \(v\). Answer: \(\sqrt{1 - v^2}\)

Step 1: Define x and use the identity

Define \(x\) as \(x = \sin^{-1}v\). The trigonometric identity is given by: $$\sin^2x + \cos^2x = 1$$ Since \(x = \sin^{-1}v\), we know that \(\sin x = v\).

Step 2: Replace \(\sin x\) in the identity

Using \(\sin x = v\), we replace \(\sin x\) in the identity: $$v^2 + \cos^2x = 1$$

Step 3: Solve for \(\cos x\)

We want to find the expression for \(\cos x\), so let's solve for \(\cos x\): $$\cos^2x = 1 - v^2$$ Now, we take the square root of both sides: $$\cos x = \sqrt{1 - v^2}$$

Step 4: Write the final algebraic expression

Now that we have found \(\cos x\), we can write the given expression as an algebraic expression in \(v\): $$\cos \left(\sin ^{-1} v\right) = \sqrt{1 - v^2}$$

Key Concepts

Understanding Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles when we know the value of a trigonometric function. For instance, if we have a value like \(v\) and know \( \sin x = v \), the inverse function \( \sin^{-1}(v) \) gives us the angle \(x\).

• These functions are essential for solving trigonometric equations where the angle is unknown.
• \( \sin^{-1}, \cos^{-1} \), and \( \tan^{-1} \) are the most common inverse functions.
• They are often used in calculus and geometry to make complex calculations simpler.

In the exercise, knowing \( \sin^{-1}(v) = x \) allows you to determine the corresponding angle for a given sine value. This angle helps find other trigonometric functions, such as cosine, by using identities.

Transforming Expressions Through Algebraic Manipulation

Algebraic expressions are combinations of numbers, variables, and operators like addition and multiplication. They are crucial for representing mathematical ideas in a concise form.

• In mathematics, we often rewrite expressions to make them simpler or solve equations.
• This involves using algebraic properties and identities.
• In our problem, we transformed a trigonometric expression into an equivalent algebraic form.

By substituting \( \sin x = v \) into the identity \( \sin^2 x + \cos^2 x = 1 \), we simplified the problem by expressing \( \cos x \) in terms of \(v\).
The process of solving for \( \cos x \) from \( \cos^2 x = 1 - v^2 \) teaches us how interconnected trigonometric identities and algebra can be.

Exploring Fundamental Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent relate angles in a triangle to the ratios of its sides. Understanding these can help you solve a variety of mathematical problems.

• Sine, cosine, and tangent are the basic trigonometric functions.
• They help us understand angles and distances in both two-dimensional and three-dimensional spaces.
• Combining these with inverse trigonometric functions provides a powerful tool for solving equations.

In this example, cosine is determined through a relationship with sine, using the identity \( \sin^2 x + \cos^2 x = 1 \). This reflects how trigonometric identities create a link between different trig functions, providing different ways to understand and calculate values. When \( \cos(\sin^{-1}(v)) = \sqrt{1 - v^2} \), it highlights the precision of these relationships in describing angles and their properties.