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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 1 - FunctionsProblema 1.3.38
Capitolo 1, Problema 1.3.38

What happens if you take B = 2π in the addition formulas? Do the results agree with something you already know?

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Consider the addition formulas for sine and cosine: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \) and \( \cos(A + B) = \cos A \cos B - \sin A \sin B \).
Substitute \( B = 2\pi \) into these formulas. Recall that \( \cos(2\pi) = 1 \) and \( \sin(2\pi) = 0 \).
For the sine addition formula, substitute \( B = 2\pi \): \( \sin(A + 2\pi) = \sin A \cdot 1 + \cos A \cdot 0 = \sin A \).
For the cosine addition formula, substitute \( B = 2\pi \): \( \cos(A + 2\pi) = \cos A \cdot 1 - \sin A \cdot 0 = \cos A \).
These results show that \( \sin(A + 2\pi) = \sin A \) and \( \cos(A + 2\pi) = \cos A \), which agree with the periodicity of sine and cosine functions, confirming that they repeat every \( 2\pi \).

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Addition Formulas

Addition formulas in trigonometry express the sine and cosine of the sum or difference of two angles. For example, the sine addition formula states that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). These formulas are essential for simplifying expressions involving trigonometric functions and for solving equations that include angles.
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Additional Rules for Indefinite Integrals

Periodic Functions

Trigonometric functions such as sine and cosine are periodic, meaning they repeat their values in regular intervals. The period of sine and cosine is 2π, which indicates that sin(x + 2π) = sin(x) and cos(x + 2π) = cos(x). Understanding periodicity is crucial when analyzing the behavior of trigonometric functions over different intervals.
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Graphs of Secant and Cosecant Functions

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 represents the cosine and the y-coordinate represents the sine of an angle. Using the unit circle helps visualize how angles relate to their sine and cosine values, especially when considering angles like B = 2π.
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Evaluate Composite Functions - Values on Unit Circle
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Composition of Functions


A balloon’s volume V is given by V = s² + 2s + 3 cm³, where s is the ambient temperature in °C. The ambient temperature s at time t minutes is given by s = 2t − 3 °C. Write the balloon’s volume V as a function of time t.

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Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.


Graph the function f (x) = sin³ x.

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Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.


Graph four periods of the function f (x) = −tan 2x.

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Finding a Viewing Window


In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.


y = 3 cos 60x

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Can a function be both even and odd? Give reasons for your answer.

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Graphing


In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.


y = (−2x)²/³

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