교과서 질문
In Exercises 63–82, use a sketch to find the exact value of each expression. cot (csc⁻¹ 8)
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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
모든 교과서
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 83
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 832장, 문제 83
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. tan (cos⁻¹ x)
검증된 단계별 안내1
Recognize that the expression \( \tan(\cos^{-1} x) \) involves the tangent of an angle whose cosine is \( x \). Let \( \theta = \cos^{-1} x \), so \( \cos \theta = x \).
Since \( \theta \) is an angle in a right triangle, draw a right triangle where the adjacent side to angle \( \theta \) is \( x \) and the hypotenuse is 1 (because cosine is adjacent over hypotenuse).
Use the Pythagorean theorem to find the length of the opposite side: \( \text{opposite} = \sqrt{1^2 - x^2} = \sqrt{1 - x^2} \).
Recall that \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). Substitute the values found: \( \tan(\cos^{-1} x) = \frac{\sqrt{1 - x^2}}{x} \).
Express the final algebraic expression for \( \tan(\cos^{-1} x) \) as \( \frac{\sqrt{1 - x^2}}{x} \), assuming \( x > 0 \) to keep the expression defined.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Inverse Trigonometric Functions
Inverse trigonometric functions, like cos⁻¹(x), return an angle whose trigonometric ratio equals x. Understanding how to interpret these functions is essential for converting expressions involving inverse trig functions into geometric or algebraic forms.
추천 영상:
4:28
Introduction to Inverse Trig Functions
Right Triangle Definitions of Trigonometric Ratios
Trigonometric ratios such as sine, cosine, and tangent can be represented as ratios of sides in a right triangle. Using a right triangle to represent an angle from an inverse trig function helps translate the problem into algebraic expressions involving side lengths.
추천 영상:
5:19Solving Right Triangles with the Pythagorean Theorem
Pythagorean Theorem
The Pythagorean theorem relates the sides of a right triangle: a² + b² = c². It is crucial for finding missing side lengths when one side and an angle are known, enabling the expression of trigonometric ratios purely in terms of x.
추천 영상:
5:19Solving Right Triangles with the Pythagorean Theorem
관련 실천
교과서 질문
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. sec (cos⁻¹ 1/x)
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교과서 질문
In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g. f(x) = 2 cos x, g(x) = cos 2x, h(x) = (f + g)(x)
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교과서 질문
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. sin (tan⁻¹ x)
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교과서 질문
In Exercises 79–82, graph f, g, and h in the same rectangular coordinate system for 0 ≤ x ≤ 2π. Obtain the graph of h by adding or subtracting the corresponding y-coordinates on the graphs of f and g. f(x) = cos x, g(x) = sin 2x, h(x) = (f − g)(x)
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교과서 질문
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)
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