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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.6.4b
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)

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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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Introduction 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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Introduction to Trigonometric Functions
Related Practice
Textbook Question

91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]

b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.

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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.

67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

Textbook Question

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

2. b. tan^(-1)(√3)

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

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

6. b. arccsc(-2/√3)