Textbook Question
What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y=x?
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.6.5c
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
5. c. arccos(√3/2)
Verified step by step guidance1
Recall that the function arccos(x) gives the angle \( \theta \) whose cosine is \( x \), and its principal value range is \( 0 \leq \theta \leq \pi \) (or 0 to 180 degrees).
Identify the value inside the arccos function: \( \sqrt{3}/2 \). Recognize this as a common cosine value from special angles in the unit circle.
Recall the reference triangle or unit circle values where \( \cos \theta = \sqrt{3}/2 \). This corresponds to an angle of \( \pi/6 \) radians (or 30 degrees) in the first quadrant.
Since arccos returns values in the range \( [0, \pi] \), consider the angle in the first quadrant where cosine is positive, which is \( \theta = \pi/6 \).
Conclude that \( \arccos(\sqrt{3}/2) = \pi/6 \) radians, based on the reference triangle and the principal value range of arccos.
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2mKey Concepts
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Inverse Trigonometric Functions
Inverse trigonometric functions, like arccos, return the angle whose trigonometric ratio matches a given value. For arccos(x), the output is the angle in [0, π] whose cosine is x. Understanding their domain and range is essential for correctly interpreting the angle values.
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Derivatives of Other Inverse Trigonometric Functions
Reference Triangles
Reference triangles are right triangles used to relate trigonometric ratios to specific angles in different quadrants. By using known side ratios, they help find angles corresponding to given trigonometric values, especially when dealing with angles beyond the first quadrant.
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Guided course
6:04Introduction to Trigonometric Functions
Quadrants and Angle Significance
The unit circle is divided into four quadrants, each with specific signs for sine, cosine, and tangent. Knowing which quadrant an angle lies in helps determine the correct angle measure from an inverse trig value, especially when multiple angles share the same cosine value.
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5:30Trig Values in Quadrants II, III, & IV
Related Practice
Textbook Question
86. This exercise explores the difference between
lim(x→∞)(1 + 1/x²)^x
and
lim(x→∞)(1 + 1/x)^x = e
c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. x²e^(-x)
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
c. lim(x→∞) sinh x
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Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
c. y = e^(-x) + Ce^(-(3/2)x)
Textbook Question
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
c. 3ln ∛(t² - 1) - ln(t+1)