Problem 79
Question
Use the graph of \(f\) to find the simplest expression \(g(x)\) such that the equation \(f(x)=g(x)\) is an Identity. Verify this identity. $$f(x)=\sec x\left(\sin x \cos x+\cos ^{2} x\right)-\sin x$$
Step-by-Step Solution
Verified Answer
The simplest expression for \( g(x) \) is \( \cos x \). It is an identity for \( f(x).\)
1Step 1: Expand f(x)
Look at the expression for \( f(x) = \sec x (\sin x \cos x + \cos^2 x) - \sin x \). Distribute \( \sec x \) across the terms inside the parentheses. This results in \( \sec x \sin x \cos x + \sec x \cos^2 x - \sin x \).
2Step 2: Simplify Using Trigonometric Identities
Recall the identities: \( \sec x = \frac{1}{\cos x} \), \( \sin x \cdot \cos x = \frac{1}{2}\sin 2x \), and \( \cos^2 x = 1 - \sin^2 x \). Substitute them into the expression: \( \frac{\sin x}{\cos x} \cdot \cos x + \frac{1}{\cos x} \cdot \cos^2 x - \sin x \). This simplifies to \( \sin x + \cos x - \sin x \).
3Step 3: Simplify Further
Simplify \( \sin x + \cos x - \sin x \) to \( \cos x \). This is the last term after cancellation.
4Step 4: Express g(x) and Verify the Identity
We found that \( g(x) = \cos x \). Substitute back to check the function: \( f(x) = \sec x (\sin x \cos x + \cos^2 x) - \sin x = \cos x = g(x) \). This confirms that the equation \( f(x) = g(x) \) is an identity.
Key Concepts
Trigonometric FunctionsSimplificationTrigonometric Equations
Trigonometric Functions
Trigonometric functions are foundational concepts in mathematics, often linked to the geometry of circles and angles. These functions include sine (\( \sin \)), cosine (\( \cos \)), and secant (\( \sec \)), among others. Each of these functions has specific roles and properties. For example, the sine function provides the ratio of the opposite side to the hypotenuse in a right triangle, while the cosine function gives the ratio of the adjacent side to the hypotenuse.
The secant function, \( \sec x \), is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This means when \( \cos x \) is zero, \( \sec x \) tends toward infinity because division by zero is undefined. Knowing these relationships allows us to manipulate and simplify trigonometric expressions effectively. Understanding trigonometric functions is essential not only in solving equations but also in analyzing periodic phenomena in real life like sound waves and radiation patterns.
The secant function, \( \sec x \), is the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This means when \( \cos x \) is zero, \( \sec x \) tends toward infinity because division by zero is undefined. Knowing these relationships allows us to manipulate and simplify trigonometric expressions effectively. Understanding trigonometric functions is essential not only in solving equations but also in analyzing periodic phenomena in real life like sound waves and radiation patterns.
Simplification
Simplification in mathematics refers to reducing an expression to its simplest form. It often involves using identities or properties to rewrite parts of the expression. For trigonometric functions, simplification can involve using basic identities like \( \cos^2 x + \sin^2 x = 1 \), or more advanced transformations like the double angle identity \( \sin(2x) = 2 \sin x \cos x \).
In our example, we simplify \( f(x) = \sec x (\sin x \cos x + \cos^2 x) - \sin x \) by first distributing \( \sec x \) across the terms inside the parenthesis. Then, utilizing basic identities, the expression is reduced step by step to \( \cos x \). Here, identifying terms that can cancel, such as \( \sin x - \sin x \), and recognizing substitutions like \( \cos^2 x = 1 - \sin^2 x \), play crucial roles in reaching the simplest expression. Practicing simplification helps in efficiently solving more complex mathematical problems.
In our example, we simplify \( f(x) = \sec x (\sin x \cos x + \cos^2 x) - \sin x \) by first distributing \( \sec x \) across the terms inside the parenthesis. Then, utilizing basic identities, the expression is reduced step by step to \( \cos x \). Here, identifying terms that can cancel, such as \( \sin x - \sin x \), and recognizing substitutions like \( \cos^2 x = 1 - \sin^2 x \), play crucial roles in reaching the simplest expression. Practicing simplification helps in efficiently solving more complex mathematical problems.
Trigonometric Equations
Trigonometric equations are equations that involve trigonometric functions. Solving these equations often requires algebraic manipulation and the use of trigonometric identities. The objective is typically to simplify the equation to a more recognizable form or to find specific values of \( x \) that satisfy the equation.
In the original exercise, we transformed \( f(x) \) into a simpler function \( g(x) \), verifying that both functions were identical. This demonstrates the process of establishing an identity, where two expressions are equivalent for all values of the variable within their domain. It's important in solving trigonometric equations to methodically use identities, such as reciprocal identities or angle sum identities, to manipulate and simplify the equation. Success in solving these equations often comes down to choosing the right identities and performing the algebraic steps in the correct order.
In the original exercise, we transformed \( f(x) \) into a simpler function \( g(x) \), verifying that both functions were identical. This demonstrates the process of establishing an identity, where two expressions are equivalent for all values of the variable within their domain. It's important in solving trigonometric equations to methodically use identities, such as reciprocal identities or angle sum identities, to manipulate and simplify the equation. Success in solving these equations often comes down to choosing the right identities and performing the algebraic steps in the correct order.
Other exercises in this chapter
Problem 78
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Use the graph of \(f\) to find the simplest expression \(g(x)\) such that the equation \(f(x)=g(x)\) is an Identity. Verify this identity. $$f(x)=\frac{\sin ^{3
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n designing a collector for solar power, an important consideration is the amount of sunlight that is transmitted through the glass into the water being heated.
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