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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 7.6.1c
Chapter 7, Problem 7.6.1c

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
1. c. tan^(-1)(1/√3)

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Identify the value inside the inverse tangent function: here it is \(\frac{1}{\sqrt{3}}\).
Recall that \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\) in a right triangle, so we want to find an angle \(\theta\) such that \(\tan \theta = \frac{1}{\sqrt{3}}\).
Recognize the common special angle where \(\tan \theta = \frac{1}{\sqrt{3}}\) is \(\theta = 30^\circ\) or \(\theta = \frac{\pi}{6}\) radians, based on the reference triangle with sides 1 (opposite), \(\sqrt{3}\) (adjacent), and 2 (hypotenuse).
Since the problem asks to use reference triangles in an appropriate quadrant, consider the principal value range of \(\tan^{-1}\), which is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), so the angle is in the first quadrant where tangent is positive.
Conclude that the angle corresponding to \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\) is the reference angle \(\frac{\pi}{6}\) radians (or \(30^\circ\)) in the first quadrant.

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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, arctan or tan⁻¹, 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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Inverse Tangent

Reference Triangles

Reference triangles are right triangles drawn in a coordinate plane to help find angles and trigonometric values in different quadrants. By using the acute angle in the triangle and the signs of trigonometric functions in each quadrant, one can determine the actual angle measure.
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Introduction to Trigonometric Functions

Quadrants and Sign of Trigonometric Functions

The coordinate plane is divided into four quadrants, each affecting the sign of sine, cosine, and tangent functions. Knowing the quadrant helps determine the correct angle corresponding to a trigonometric value, especially when using inverse functions that have restricted ranges.
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Introduction 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:


c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).


70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

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

c. x=3

Textbook Question

9. True, or false? As x→∞,

c. x = O(x+5)

1
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Textbook Question

In Exercises 41–44:


c. Evaluate df/dx at x = a and df⁻¹/dx at x = f(a) to show that

(df⁻¹/dx)|ₓ₌f(a) = 1 / (df/dx)|ₓ₌a


44. f(x) = 2x², x ≥ 0, a = 5

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)).

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

Textbook Question

c. Find the slopes of the tangent lines to the graphs of h and k at (2, 2) and (−2, −2).