Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problema 66

Capitolo 2, Problema 66

In Exercises 61–66, use the method of adding y-coordinates to graph each function for 0 ≤ x ≤ 2π. y = cos x + sin 2x

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Identify the two functions involved: \(y_1 = \cos x\) and \(y_2 = \sin 2x\). We will graph each separately over the interval \(0 \leq x \leq 2\pi\).

Create a table of values for \(y_1 = \cos x\) by choosing key points in the interval \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\) and calculate \(\cos x\) at these points.

Similarly, create a table of values for \(y_2 = \sin 2x\) at the same key points by calculating \(\sin 2x\) for each \(x\) value.

Add the corresponding \(y\)-coordinates from the two tables to find the values of \(y = \cos x + \sin 2x\) at each key point. This means for each \(x\), compute \(y = y_1 + y_2\).

Plot the points \((x, y)\) obtained from the sums on the coordinate plane and connect them smoothly to graph the function \(y = \cos x + \sin 2x\) over the interval \(0 \leq x \leq 2\pi\).

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Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their values over a specified interval, typically one or more periods. Understanding the shape, amplitude, period, and phase shift of sine and cosine functions helps in accurately sketching their graphs.

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Introduction to Trigonometric Functions

Sum of Functions and Pointwise Addition

When adding two functions, the resulting function's value at each x is the sum of the individual function values at that x. For trigonometric functions, this means adding the y-coordinates of each function point-by-point to obtain the combined graph.

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Properties of Sine and Cosine Functions

Sine and cosine functions have specific properties such as amplitude, period, and frequency. For example, sin(2x) has twice the frequency of sin(x), resulting in a shorter period. Recognizing these properties is essential for understanding how the combined function behaves.

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