Textbook Question
5. e^(2t)-3e^t = 0
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.6.11
Find the values in Exercises 9–12.
11. tan(arcsin(-1/2))
Verified step by step guidance1
Recognize that the expression is \( \tan(\arcsin(-\frac{1}{2})) \). Here, \( \arcsin(-\frac{1}{2}) \) represents an angle \( \theta \) such that \( \sin(\theta) = -\frac{1}{2} \).
Let \( \theta = \arcsin(-\frac{1}{2}) \). Since \( \sin(\theta) = -\frac{1}{2} \), we want to find \( \tan(\theta) \). Recall that \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
Use the Pythagorean identity to find \( \cos(\theta) \): \( \cos(\theta) = \pm \sqrt{1 - \sin^2(\theta)} = \pm \sqrt{1 - \left(-\frac{1}{2}\right)^2} = \pm \sqrt{1 - \frac{1}{4}} = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \).
Determine the correct sign of \( \cos(\theta) \) by considering the range of \( \arcsin \), which is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). In this interval, cosine is positive for angles where sine is negative, so \( \cos(\theta) = \frac{\sqrt{3}}{2} \).
Finally, compute \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} \).
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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 arcsin, return the angle whose trigonometric ratio equals a given value. For example, arcsin(-1/2) gives the angle θ in the range [-π/2, π/2] such that sin(θ) = -1/2.
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Derivatives of Other Inverse Trigonometric Functions
Relationship Between Sine and Tangent
To find tan(arcsin(x)), use the identity tan(θ) = sin(θ)/cos(θ). Knowing sin(θ), you can find cos(θ) using the Pythagorean identity cos²(θ) = 1 - sin²(θ), then compute the tangent.
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Pythagorean Identity
The Pythagorean identity states that sin²(θ) + cos²(θ) = 1 for any angle θ. This allows calculation of one trigonometric function if the other is known, essential for evaluating expressions like tan(arcsin(x)).
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Related Practice
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Textbook Question
143.
b. Find the average value of ln(x) over [1, e].