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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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 91
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 912장, 문제 91
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 (sin⁻¹ x/√x²+4)
검증된 단계별 안내1
Identify the angle \( \theta \) such that \( \theta = \sin^{-1} \left( \frac{x}{\sqrt{x^2 + 4}} \right) \). This means \( \sin \theta = \frac{x}{\sqrt{x^2 + 4}} \).
Draw a right triangle where the angle \( \theta \) has an opposite side of length \( x \) and a hypotenuse of length \( \sqrt{x^2 + 4} \) based on the sine definition \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
Use the Pythagorean theorem to find the adjacent side of the triangle: \( \text{adjacent} = \sqrt{(\text{hypotenuse})^2 - (\text{opposite})^2} = \sqrt{(\sqrt{x^2 + 4})^2 - x^2} \).
Simplify the adjacent side expression: \( \sqrt{x^2 + 4 - x^2} = \sqrt{4} = 2 \).
Recall that \( \sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} \). Substitute the known values to write \( \sec (\sin^{-1} (\frac{x}{\sqrt{x^2 + 4}})) = \frac{\sqrt{x^2 + 4}}{2} \) as the algebraic expression.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Inverse Trigonometric Functions
Inverse trigonometric functions, like sin⁻¹ (arcsin), return the angle whose trigonometric ratio equals a given value. Understanding their domain and range is essential, as they help translate between angle measures and ratio values, especially when constructing right triangles.
추천 영상:
4:28
Introduction to Inverse Trig Functions
Right Triangle Trigonometry
Right triangle trigonometry relates the sides and angles of a right triangle using ratios such as sine, cosine, and secant. By interpreting inverse trig expressions as angles in a triangle, one can find unknown sides or other trigonometric values algebraically.
추천 영상:
04:3945-45-90 Triangles
Algebraic Manipulation of Trigonometric Expressions
Converting trigonometric expressions involving inverse functions into algebraic forms requires substituting sides of a triangle and simplifying. This often involves using the Pythagorean theorem and expressing trigonometric ratios in terms of the variable x.
추천 영상:
6:36Simplifying Trig Expressions
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. csc (cot⁻¹ x)
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교과서 질문
The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. f(x) = sin⁻¹ x + π/2
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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
The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. f(x) = cos⁻¹ (x + 1)
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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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