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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 94
Chapter 2, Problem 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)

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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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Solving 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.
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Graphs of Secant and Cosecant Functions
Related Practice
Textbook Question

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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Textbook Question

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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Textbook Question

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)

749
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Textbook Question

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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Textbook Question

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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Textbook Question

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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