Question

Trigonometric ldentities Let f(x)=\sin x+\cos x (a) Graph \(y=f(x)\) . Describe the graph. (b) Use the graph to identify the amplitude, period, horizontal shift, and vertical shift. (c) Use the formula \(\sin \alpha \cos \beta+\cos \alpha \sin \beta=\sin (\alpha+\beta)\) for the sine of the sum of two angles to confirm your answers.

Short answer

The graph of the function \(y=f(x)=\sin(x)+\cos(x)\) has an amplitude of \(\sqrt{2}\), a period of \(2\pi\), a horizontal shift of \(-\pi/4\), and no vertical shift. These are confirmed by the trigonometric identity \(\sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta) = \sin(\alpha + \beta)\).

Step-by-step solution

Graphing the function

First, graph the function \(y=f(x)=\sin(x)+\cos(x)\) on any graphing software or calculator. Make sure to mark the points where the function reaches its highest and lowest points, and the points where it intersects with the x-axis.

Identifying Key Graph Features

Next, identify the key features of the graph based on visual observation. The amplitude (maximum value - minimum value) / 2, the period is the distance covered in one full cycle of the wave, and note any horizontal and vertical shifts (the graph's deviation from standard sine or cosine function).

Confirming Observations using Trigonometric Identity

Use the identity \(\sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta) = \sin(\alpha + \beta)\) to confirm the graph's features. Here, set \(\alpha = \beta = x\). Then \(\sin(x) \cos(x) + \cos(x) \sin(x) = 2 \sin(x) \cos(x) = \sqrt{2} \sin(x + \pi/4)\). The \(\sqrt{2}\) confirms the amplitude observed and the term \(\sin(x + \pi/4)\) confirms the graph's period and shift.