Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
69. ∫(from 5/4 to 2)dx/(1-x²)
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 7.6.8b
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
8. b. arccot(√3)
Verified step by step guidance1
Recall that \( \arccot(x) \) is the angle \( \theta \) such that \( \cot(\theta) = x \). Here, we want to find \( \theta = \arccot(\sqrt{3}) \).
Use the definition of cotangent in terms of sine and cosine: \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). So, we need to find an angle \( \theta \) where \( \frac{\cos(\theta)}{\sin(\theta)} = \sqrt{3} \).
Recognize that \( \cot(\theta) = \sqrt{3} \) corresponds to a reference triangle where the adjacent side is \( \sqrt{3} \) and the opposite side is 1. This is a 30°-60°-90° triangle, where \( \cot(30^\circ) = \sqrt{3} \).
Determine the quadrant for \( \arccot(\sqrt{3}) \). Since \( \sqrt{3} > 0 \), \( \theta \) lies in the first quadrant where cotangent is positive.
Conclude that \( \arccot(\sqrt{3}) \) is the angle in the first quadrant with reference angle 30°, so \( \theta = 30^\circ \) or in radians \( \theta = \frac{\pi}{6} \).
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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 arccot, return the angle whose trigonometric ratio equals a given value. For arccot(√3), we seek an angle whose cotangent is √3. Understanding their ranges and outputs is essential for correctly identifying the angle.
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06:35
Derivatives of Other Inverse Trigonometric Functions
Reference Triangles
Reference triangles are right triangles used to relate trigonometric ratios to specific angles. By constructing a triangle with sides matching the given ratio, we can find the acute reference angle, which helps determine the actual angle in the correct quadrant.
Recommended video:
6:04Introduction to Trigonometric Functions
Quadrants and Angle Sign Conventions
The value of trigonometric functions varies by quadrant, affecting the angle's sign and measure. Knowing which quadrant the angle lies in, based on the function and its value, is crucial to correctly interpreting the inverse function's output.
Recommended video:
5:30Trig Values in Quadrants II, III, & IV
Related Practice
Textbook Question
132. Let f(x) = e^x / (1 + e^(2x)).
b. Find all inflection points for f.
Textbook Question
In Exercises 1–4, solve for t.
2. b. e^(kt) = 10
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.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
Textbook Question
Verify the integration formulas in Exercises 37–40.
37. b. ∫sech(x)dx = sin⁻¹(tanh x) + C
1
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Textbook Question
131. Let f(x) = x * e^(−x).
b. Find all inflection points for f.