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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 1 - FunctionsProblem 1.3.44
Chapter 1, Problem 1.3.44

Evaluate cos (11π/12) as cos (π/4 + 2π/3).

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Recognize that the expression cos(11π/12) can be rewritten using the angle addition formula for cosine: cos(π/4 + 2π/3).
Recall the angle addition formula for cosine: cos(A + B) = cos(A)cos(B) - sin(A)sin(B).
Substitute A = π/4 and B = 2π/3 into the angle addition formula: cos(π/4 + 2π/3) = cos(π/4)cos(2π/3) - sin(π/4)sin(2π/3).
Find the values of cos(π/4), cos(2π/3), sin(π/4), and sin(2π/3) using known trigonometric values: cos(π/4) = √2/2, cos(2π/3) = -1/2, sin(π/4) = √2/2, sin(2π/3) = √3/2.
Substitute these values into the formula: cos(π/4 + 2π/3) = (√2/2)(-1/2) - (√2/2)(√3/2).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is periodic with a period of 2π, meaning that cos(θ) = cos(θ + 2πn) for any integer n. Understanding the properties of the cosine function is essential for evaluating expressions involving angles.
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Graph of Sine and Cosine Function

Angle Addition Formula

The angle addition formula for cosine states that cos(a + b) = cos(a)cos(b) - sin(a)sin(b). This formula allows us to break down the cosine of a sum of angles into simpler components, making it easier to evaluate complex expressions. It is particularly useful when dealing with angles that are not standard, such as π/4 and 2π/3.
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Additional Rules for Indefinite Integrals

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It provides a geometric interpretation of trigonometric functions, where the x-coordinate of a point on the circle corresponds to the cosine of the angle formed with the positive x-axis. Familiarity with the unit circle helps in determining the values of trigonometric functions for various angles, including those expressed in radians.
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Related Practice
Textbook Question

[Technology Exercise]


a. Graph the functions f(x) = 3/(x − 1) and g(x) = 2/(x + 1) together to identify the values of x for which


3/(x − 1) < 2/(x + 1)


b. Confirm your findings in part (a) algebraically.

Textbook Question

In Exercises 19–32, find the (a) domain and (b) range.


𝔂 = |x| - 2

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Textbook Question

Even and Odd Functions


In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.


h(t) = |t³|

Textbook Question

Increasing and Decreasing Functions


Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.


y = 1/|x|

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Textbook Question

Theory and Examples


The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.

Textbook Question

Using the Half-Angle Formulas


Find the function values in Exercises 47–50.


cos² 5π/12