In Exercises 83–94, 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. cos (sin⁻¹ 1/x)

검증된 단계별 안내

Recognize that the expression is \( \cos(\sin^{-1}(1/x)) \). Let \( \theta = \sin^{-1}(1/x) \), which means \( \sin(\theta) = \frac{1}{x} \).

Since \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), represent this as a right triangle where the opposite side is 1 and the hypotenuse is \( x \).

Use the Pythagorean theorem to find the adjacent side of the triangle: \( \text{adjacent} = \sqrt{x^2 - 1^2} = \sqrt{x^2 - 1} \).

Recall that \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \). Substitute the values from the triangle to get \( \cos(\sin^{-1}(1/x)) = \frac{\sqrt{x^2 - 1}}{x} \).

Write the final algebraic expression for the original expression as \( \frac{\sqrt{x^2 - 1}}{x} \), assuming \( x > 0 \) to keep the square root defined and positive.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Inverse Trigonometric Functions

Inverse trigonometric functions, like sin⁻¹(x), return the angle whose trigonometric ratio equals x. Understanding that sin⁻¹(1/x) gives an angle θ such that sin(θ) = 1/x is essential for rewriting expressions involving these functions.

Right Triangle Definitions of Trigonometric Ratios

Trigonometric ratios (sine, cosine, tangent) can be represented as ratios of sides in a right triangle. By constructing a right triangle with an angle θ where sin(θ) = opposite/hypotenuse = 1/x, we can find other ratios like cos(θ) in terms of x.

Pythagorean Theorem

The Pythagorean theorem relates the sides of a right triangle: (hypotenuse)² = (opposite)² + (adjacent)². This allows us to find the missing side length when two sides are known, which is crucial for expressing cos(sin⁻¹(1/x)) algebraically.