Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)
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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 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 7.6.4b
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
4. b. arcsin(-1/√2)
Verified step by step guidance1
Recall that the function \(\arcsin(x)\) gives the angle \(\theta\) whose sine is \(x\), with the range of \(\arcsin\) restricted to \([-\frac{\pi}{2}, \frac{\pi}{2}]\) (Quadrants IV and I).
Identify the value inside the \(\arcsin\): here it is \(-\frac{1}{\sqrt{2}}\). Recognize that \(\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), so the reference angle is \(\frac{\pi}{4}\).
Since the sine value is negative, and \(\arcsin\) outputs angles in \([-\frac{\pi}{2}, \frac{\pi}{2}]\), the angle must be in Quadrant IV where sine is negative.
Use the reference triangle in Quadrant IV to express the angle as \(-\frac{\pi}{4}\), because sine of \(-\frac{\pi}{4}\) is \(-\frac{1}{\sqrt{2}}\).
Therefore, the angle \(\theta = \arcsin\left(-\frac{1}{\sqrt{2}}\right)\) corresponds to \(-\frac{\pi}{4}\) within the principal range of \(\arcsin\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Sine Function (arcsin)
The inverse sine function, arcsin, returns the angle whose sine value is a given number. Its range is limited to angles between -π/2 and π/2 (or -90° to 90°), meaning it outputs angles in the first and fourth quadrants. Understanding this helps identify the principal value of the angle.
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Inverse Sine
Reference Triangles
Reference triangles are right triangles used to find angle measures based on known trigonometric ratios. By considering the absolute value of the sine and the quadrant of the angle, you can determine the exact angle measure and its sign, aiding in solving inverse trig problems.
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6:04Introduction to Trigonometric Functions
Quadrants and Sign of Trigonometric Functions
The sign of sine depends on the quadrant where the angle lies: sine is positive in the first and second quadrants and negative in the third and fourth. Knowing the quadrant helps determine the correct angle corresponding to a given sine value, especially when the value is negative.
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6:04Introduction to Trigonometric Functions
Related Practice
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.
70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2
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Textbook Question
75. b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
g(x) = x(ln x)²
Textbook Question
3. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
b. ln(3x² - 9x) + ln(1/3x)
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:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
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