{"id":1903,"date":"2025-07-30T21:17:28","date_gmt":"2025-07-30T21:17:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1903"},"modified":"2025-08-13T03:07:45","modified_gmt":"2025-08-13T03:07:45","slug":"graphs-of-the-sine-and-cosine-function-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-sine-and-cosine-function-learn-it-2\/","title":{"raw":"Graphs of the Sine and Cosine Function: Learn It 2","rendered":"Graphs of the Sine and Cosine Function: Learn It 2"},"content":{"raw":"

Period and Amplitude<\/h2>\r\nAs we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function<\/strong> is known as a sinusoidal function<\/strong>.\r\n\r\n
\r\n

general form of sine and cosine<\/h3>\r\nThe general forms of sinusoidal functions are\r\n
[latex]y = A\\sin (Bx\u2212C) + D[\/latex]<\/div>\r\nand\r\n
[latex]y = A\\cos (Bx\u2212C) + D[\/latex]<\/div>\r\n<\/section>\r\n

Determining the Period of Sinusoidal Functions<\/h3>\r\nLooking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.\r\n\r\nIn the general formula, B<\/em> is related to the period by [latex]P=\\frac{2\u03c0}{|B|}[\/latex]. If [latex]|B| > 1[\/latex], then the period is less than [latex]2\u03c0[\/latex] and the function undergoes a horizontal compression, whereas if [latex]|B| < 1[\/latex], then the period is greater than [latex]2\u03c0[\/latex] and the function undergoes a horizontal stretch. For example, [latex]f(x) = \\sin(x), B= 1[\/latex], so the period is [latex]2\u03c0[\/latex], which we knew. If [latex]f(x) =\\sin (2x)[\/latex], then [latex]B= 2[\/latex], so the period is [latex]\u03c0[\/latex] and the graph is compressed. If [latex]f(x) = \\sin\\left(\\frac{x}{2} \\right)[\/latex], then [latex]B=\\frac{1}{2}[\/latex], so the period is [latex]4\u03c0[\/latex] and the graph is stretched. Notice in the figure how period is indirectly related to [latex]|B|[\/latex].\r\n
\"A<\/figure>\r\n
\r\n

period of sinusoidal functions<\/h3>\r\nFor a sinusoidal function in the form [latex]y = A\\sin (Bx\u2212C) + D[\/latex] or [latex]y = A\\cos (Bx\u2212C) + D[\/latex], the period is [latex]\\frac{2\u03c0}{|B|}[\/latex].\r\n\r\n<\/section>
Determine the period of the function [latex]f(x) = \\sin\\left(\\frac{\u03c0}{6}x\\right)[\/latex].[reveal-answer q=\"616023\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"616023\"]Let's begin by comparing the equation to the general form [latex]y=A\\sin(Bx)[\/latex].In the given equation, [latex]B =\\frac{\u03c0}{6}[\/latex], so the period will be\r\n

[latex]\\begin{align}P&=\\frac{\\frac{2}{\\pi}}{|B|} \\\\ &=\\frac{2\\pi}{\\frac{x}{6}} \\\\ &=2\\pi\\times \\frac{6}{\\pi} \\\\ &=12 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

Determine the period of the function [latex]g(x)=\\cos\\left(\\frac{x}{3}\\right)[\/latex].[reveal-answer q=\"111765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"111765\"][latex]6 \\pi[\/latex][\/hidden-answer]<\/section>\r\n

Determining Amplitude<\/h3>\r\nReturning to the general formula for a sinusoidal function, we have analyzed how the variable B<\/em> relates to the period. Now let\u2019s turn to the variable A<\/em> so we can analyze how it is related to the amplitude<\/strong>, or greatest distance from rest. A<\/em> represents the vertical stretch factor, and its absolute value [latex]|A|[\/latex] is the amplitude. The local maxima will be a distance [latex]|A|[\/latex] above the vertical midline<\/strong> of the graph, which is the line [latex]x = D[\/latex]; because [latex]D = 0[\/latex] in this case, the midline is the [latex]x[\/latex]-axis. The local minima will be the same distance below the midline. If [latex]|A| > 1[\/latex], the function is stretched. For example, the amplitude of [latex]f(x)=4\\sin\\left(x\\right)[\/latex] is twice the amplitude of [latex]f(x)=2\\sin\\left(x\\right)[\/latex]\r\n\r\nIf [latex]|A| < 1[\/latex], the function is compressed. The graph below compares several sine functions with different amplitudes.\r\n
\"A<\/figure>\r\n
\r\n

amplitude of sinusoidal functions<\/h3>\r\nFor a sinusoidal function in the form [latex]y = A\\sin (Bx\u2212C) + D[\/latex] or [latex]y = A\\cos (Bx\u2212C) + D[\/latex], the amplitude<\/strong> is |A|. <\/span>\r\n\r\n \r\n\r\nIn addition, notice in the example that<\/span>\r\n

[latex]|A|=\\text{amplitude}=\\frac{1}{2}|\\text{maximum}\u2212\\text{minimum}|[\/latex]<\/span><\/p>\r\n\r\n<\/section>

What is the amplitude of the sinusoidal function\u00a0[latex]f(x)=\u22124\\sin(x)[\/latex]? Is the function stretched or compressed vertically?[reveal-answer q=\"207317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"207317\"]Let\u2019s begin by comparing the function to the simplified form [latex]y=A\\sin(Bx)[\/latex].In the given function, A\u00a0<\/em>= \u22124, so the amplitude is |A<\/em>|=|\u22124| = 4. The function is stretched.\r\n

Analysis of the Solution<\/h4>\r\nThe negative value of A<\/em> results in a reflection across the x<\/em>-axis of the sine function<\/strong>.\r\n
\"A<\/figure>\r\n[\/hidden-answer]\r\n\r\n<\/section>
What is the amplitude of the sinusoidal function [latex]f(x)=\\frac{1}{2}\\sin (x)[\/latex]? Is the function stretched or compressed vertically?[reveal-answer q=\"525586\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"525586\"][latex]\\frac{1}{2}[\/latex] compressed[\/hidden-answer]<\/section>","rendered":"

Period and Amplitude<\/h2>\n

As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function<\/strong> is known as a sinusoidal function<\/strong>.<\/p>\n

\n

general form of sine and cosine<\/h3>\n

The general forms of sinusoidal functions are<\/p>\n

[latex]y = A\\sin (Bx\u2212C) + D[\/latex]<\/div>\n

and<\/p>\n

[latex]y = A\\cos (Bx\u2212C) + D[\/latex]<\/div>\n<\/section>\n

Determining the Period of Sinusoidal Functions<\/h3>\n

Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.<\/p>\n

In the general formula, B<\/em> is related to the period by [latex]P=\\frac{2\u03c0}{|B|}[\/latex]. If [latex]|B| > 1[\/latex], then the period is less than [latex]2\u03c0[\/latex] and the function undergoes a horizontal compression, whereas if [latex]|B| < 1[\/latex], then the period is greater than [latex]2\u03c0[\/latex] and the function undergoes a horizontal stretch. For example, [latex]f(x) = \\sin(x), B= 1[\/latex], so the period is [latex]2\u03c0[\/latex], which we knew. If [latex]f(x) =\\sin (2x)[\/latex], then [latex]B= 2[\/latex], so the period is [latex]\u03c0[\/latex] and the graph is compressed. If [latex]f(x) = \\sin\\left(\\frac{x}{2} \\right)[\/latex], then [latex]B=\\frac{1}{2}[\/latex], so the period is [latex]4\u03c0[\/latex] and the graph is stretched. Notice in the figure how period is indirectly related to [latex]|B|[\/latex].\n\n\n

\"A<\/figure>\n
\n

period of sinusoidal functions<\/h3>\n

For a sinusoidal function in the form [latex]y = A\\sin (Bx\u2212C) + D[\/latex] or [latex]y = A\\cos (Bx\u2212C) + D[\/latex], the period is [latex]\\frac{2\u03c0}{|B|}[\/latex].<\/p>\n<\/section>\n

Determine the period of the function [latex]f(x) = \\sin\\left(\\frac{\u03c0}{6}x\\right)[\/latex].<\/p>\n