57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If x = 4 tanθ, then cscθ = 4/x.

검증된 단계별 안내

Step 1: Begin by recalling the definitions of trigonometric functions. The tangent function is defined as \( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \), and the cosecant function is defined as \( \csc\theta = \frac{1}{\sin\theta} \).

Step 2: From the problem, we are given \( x = 4 \tan\theta \). Rearrange this equation to express \( \tan\theta \) in terms of \( x \): \( \tan\theta = \frac{x}{4} \).

Step 3: Use the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \) to relate \( \tan\theta \) to \( \sin\theta \) and \( \cos\theta \). Recall that \( \tan\theta = \frac{\sin\theta}{\cos\theta} \), so \( \sin\theta = \tan\theta \cdot \cos\theta \).

Step 4: Substitute \( \tan\theta = \frac{x}{4} \) into the expression for \( \sin\theta \). To find \( \csc\theta \), take the reciprocal of \( \sin\theta \): \( \csc\theta = \frac{1}{\sin\theta} \). Verify whether \( \csc\theta = \frac{4}{x} \) holds true by substituting \( \sin\theta \) and simplifying.

Step 5: Analyze the result. If \( \csc\theta \neq \frac{4}{x} \), provide a counterexample to show why the statement is false. If \( \csc\theta = \frac{4}{x} \), explain why the statement is true based on the trigonometric relationships derived.

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Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is essential for manipulating and simplifying trigonometric expressions, which is crucial for solving problems in calculus.

Reciprocal Functions

Reciprocal functions are pairs of trigonometric functions that are defined as the reciprocal of each other. For example, the cosecant function (cscθ) is the reciprocal of the sine function (sinθ), meaning cscθ = 1/sinθ. Recognizing these relationships helps in transforming and solving trigonometric equations, particularly when verifying statements involving these functions.

Substitution in Trigonometry

Substitution in trigonometry involves replacing one variable with another to simplify expressions or solve equations. In this context, substituting x = 4 tanθ allows us to express other trigonometric functions in terms of x. This technique is vital for analyzing relationships between different trigonometric functions and verifying the truth of statements involving them.

57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. If x = 4 tanθ, then cscθ = 4/x.