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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 7.6.1a
Chapter 7, Problem 7.6.1a

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

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1
Recall that \( \arctan(x) \) is the inverse tangent function, which gives the angle \( \theta \) whose tangent is \( x \). So, we want to find \( \theta \) such that \( \tan(\theta) = 1 \).
Identify the reference angle by considering the positive value 1 for tangent. Since \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \), a ratio of 1 means the opposite and adjacent sides of the reference triangle are equal.
Recognize that the angle with tangent 1 in the first quadrant is \( \frac{\pi}{4} \) radians (or 45 degrees), because in a right triangle with equal legs, the angles opposite those legs are equal.
Since the problem does not specify a quadrant other than the principal value range of \( \arctan \), which is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), the angle is simply the reference angle \( \frac{\pi}{4} \).
Therefore, the solution is \( \theta = \arctan(1) = \frac{\pi}{4} \), using the reference triangle 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 Trigonometric Functions

Inverse trigonometric functions, like arctan, return the angle whose trigonometric ratio equals a given value. For example, arctan(1) gives the angle whose tangent is 1. Understanding their ranges and outputs is essential for finding correct angle measures.
Recommended video:
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Derivatives of Other Inverse Trigonometric Functions

Reference Triangles

Reference triangles are right triangles used to relate trigonometric ratios to angles in different quadrants. By considering the signs of sides and angles, they help determine the actual angle corresponding to a trigonometric value in any quadrant.
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Introduction to Trigonometric Functions

Quadrants and Angle Sign Conventions

The coordinate plane is divided into four quadrants, each with specific sign rules for sine, cosine, and tangent. Knowing which quadrant an angle lies in helps assign the correct sign and value to the angle found using inverse trig functions.
Recommended video:
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Trig Values in Quadrants II, III, & IV
Related Practice
Textbook Question

Evaluate the integrals in Exercises 67–74 in terms of

a. inverse hyperbolic functions.

71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))

Textbook Question

In Exercises 1–4, solve for t.

1. a. e^(-0.3t) = 27

Textbook Question

153. The linearization of 2ˣ

a. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.

Textbook Question

Evaluate the integrals in Exercises 67–74 in terms of

a. inverse hyperbolic functions.

73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)

Textbook Question

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

a. x² + 4x

Textbook Question

9. True, or false? As x→∞,

a. x = o(x)

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