Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))
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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 7.6.2b
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
2. b. tan^(-1)(√3)
Verified step by step guidance1
Recognize that the problem asks for the angle whose tangent value is \( \sqrt{3} \), which means we want to find \( \theta = \tan^{-1}(\sqrt{3}) \).
Recall the definition of tangent in a right triangle: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). Here, the ratio is \( \sqrt{3} \) to 1.
Construct a reference triangle where the side opposite the angle \( \theta \) is \( \sqrt{3} \) and the adjacent side is 1. Use the Pythagorean theorem to find the hypotenuse: \( \text{hypotenuse} = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2 \).
Identify the angle \( \theta \) in the first quadrant (since inverse tangent outputs angles between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \)) that corresponds to this triangle. This angle is commonly known from special triangles.
Conclude that the angle \( \theta = \tan^{-1}(\sqrt{3}) \) corresponds to the angle in the reference triangle with opposite side \( \sqrt{3} \), adjacent side 1, and hypotenuse 2, which is a standard angle in trigonometry.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Tangent Function (arctan)
The inverse tangent function, denoted as tan⁻¹ or arctan, returns the angle whose tangent is a given value. It is used to find an angle when the ratio of the opposite side to the adjacent side in a right triangle is known. The output angle is typically in the range (-π/2, π/2) or (-90°, 90°).
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3:17
Inverse Tangent
Reference Triangles
Reference triangles are right triangles drawn in specific quadrants to help determine the exact angle corresponding to a trigonometric value. They use known side ratios to find angles and adjust for the quadrant by considering the sign of the trigonometric function. This method aids in finding angles beyond the principal range.
Recommended video:
6:04Introduction to Trigonometric Functions
Quadrants and Angle Sign Conventions
The coordinate plane is divided into four quadrants, each with specific sign conventions for sine, cosine, and tangent. Understanding which quadrant an angle lies in helps determine the correct angle measure and sign of the trigonometric function. For example, tangent is positive in the first and third quadrants.
Recommended video:
5:30Trig Values in Quadrants II, III, & IV
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
4. b. arcsin(-1/√2)
Textbook Question
1. Express the following logarithms in terms of ln 2 and ln 3.
b. ln(4/9)
Textbook Question
Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.
Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at
b. x=2
Textbook Question
Find the volumes of the solids in Exercises 135 and 136.
135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis are
b. vertical squares whose base edges run from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).
Textbook Question
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2