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 94
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions문제 942장, 문제 94
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)
검증된 단계별 안내1
Recognize that the expression is \( \csc(\cot^{-1} x) \). Let \( \theta = \cot^{-1} x \), which means \( \cot \theta = x \).
Recall that \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \). Since \( x > 0 \), we can represent the right triangle with adjacent side = \( x \) and opposite side = 1.
Use the Pythagorean theorem to find the hypotenuse: \( \text{hypotenuse} = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \).
Recall that \( \csc \theta = \frac{\text{hypotenuse}}{\text{opposite}} \). Substitute the values from the triangle: \( \csc \theta = \frac{\sqrt{x^2 + 1}}{1} \).
Therefore, \( \csc(\cot^{-1} x) = \sqrt{x^2 + 1} \), which is the algebraic expression in terms of \( x \).
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주요 개념
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Inverse Trigonometric Functions
Inverse trigonometric functions, like cot⁻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, cotangent, and cosecant can be represented as ratios of sides in a right triangle. Using a right triangle helps visualize and rewrite expressions like csc(cot⁻¹ x) in terms of side lengths and algebraic expressions.
추천 영상:
5:19Solving Right Triangles with the Pythagorean Theorem
Relationship Between Cotangent and Cosecant
Cotangent is the ratio of adjacent to opposite sides, while cosecant is the reciprocal of sine (hypotenuse over opposite). Understanding how to express csc(θ) when θ = cot⁻¹(x) involves using the Pythagorean theorem to find the hypotenuse and then forming the correct ratio.
추천 영상:
6:22Graphs of Secant and Cosecant Functions
관련 실천
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
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
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. h(x) = −2 tan⁻¹ 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. ___ sec (sin⁻¹ x/√x²+4)
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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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