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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Capitolo 2, Problema 88

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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Recognize that the expression is \( \sec(\cos^{-1}(1/x)) \). Let \( \theta = \cos^{-1}(1/x) \), which means \( \cos(\theta) = \frac{1}{x} \).
Since \( \theta \) is an angle in a right triangle, draw a right triangle where the adjacent side to \( \theta \) is 1 and the hypotenuse is \( x \) (because \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)).
Use the Pythagorean theorem to find the length of the opposite side: \( \text{opposite} = \sqrt{x^2 - 1^2} = \sqrt{x^2 - 1} \).
Recall that \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}} \). Using the triangle, \( \sec(\theta) = \frac{x}{1} = x \).
Therefore, \( \sec(\cos^{-1}(1/x)) \) can be expressed algebraically as \( x \).

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Inverse Trigonometric Functions

Inverse trigonometric functions, like cos⁻¹ (arccos), return an angle whose trigonometric ratio matches the given value. For cos⁻¹(1/x), the output is an angle θ such that cos(θ) = 1/x, with θ typically in the range [0, π]. Understanding this helps translate the expression into a geometric context.
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Introduction to Inverse Trig Functions

Right Triangle Definitions of Trigonometric Ratios

Trigonometric ratios such as secant are defined using right triangles: sec(θ) = hypotenuse/adjacent. By interpreting θ = cos⁻¹(1/x), we can construct a right triangle where the adjacent side is 1 and the hypotenuse is x, allowing us to express sec(θ) algebraically in terms of x.
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Solving Right Triangles with the Pythagorean Theorem

Algebraic Manipulation Using the Pythagorean Theorem

To find the missing side of the triangle, use the Pythagorean theorem: hypotenuse² = adjacent² + opposite². Given hypotenuse = x and adjacent = 1, the opposite side is √(x² - 1). This enables expressing sec(θ) = hypotenuse/adjacent = x/1 = x, or other related expressions, purely algebraically.
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