Domanda del libro di testo
In Exercises 29–44, graph two periods of the given cosecant or secant function. y = 2 sec(x + π)
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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 45
Blitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblema 45Capitolo 2, Problema 45
In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan (tan⁻¹ 125)
Guida verificata passo dopo passo1
Recognize that the expression is \( \tan(\tan^{-1}(125)) \). Here, \( \tan^{-1} \) is the inverse tangent function, also called arctangent, which returns an angle whose tangent is the given number.
Understand that \( \tan^{-1}(125) \) gives an angle \( \theta \) such that \( \tan(\theta) = 125 \).
Since \( \tan \) and \( \tan^{-1} \) are inverse functions, applying \( \tan \) to \( \tan^{-1}(125) \) essentially returns the original input value, provided the angle is within the principal range of \( \tan^{-1} \).
Therefore, the expression simplifies to \( 125 \) because \( \tan(\tan^{-1}(x)) = x \) for all real numbers \( x \).
No further calculation is needed, and the exact value of the expression is \( 125 \).
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Inverse Trigonometric Functions
Inverse trigonometric functions, like tan⁻¹ (arctan), return the angle whose trigonometric ratio equals a given value. For example, tan⁻¹(125) gives the angle whose tangent is 125. Understanding this helps in simplifying expressions involving inverse functions.
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4:28
Introduction to Inverse Trig Functions
Tangent Function and Its Properties
The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. It is periodic and defined for all real numbers except odd multiples of 90°. Knowing that tan and tan⁻¹ are inverse functions helps simplify tan(tan⁻¹(x)) to x within the function's domain.
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5:43Introduction to Tangent Graph
Simplification of Composite Functions
When a function and its inverse are composed, such as tan(tan⁻¹(x)), the result simplifies to x, provided x is within the domain of the inverse function. This principle allows direct evaluation of expressions without a calculator by recognizing the cancellation effect.
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3:48Evaluate Composite Functions - Special Cases
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In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 3 cos(2x − π)
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Domanda del libro di testo
Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = cos(x + π/2)
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In Exercises 29–51, find the exact value of each expression. Do not use a calculator. sin(cos⁻¹ 3/5)
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In Exercises 43–52, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = cos(x − π/2)
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Domanda del libro di testo
Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = 4 cos(2x − π)
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