{"id":246,"date":"2017-08-23T10:31:51","date_gmt":"2017-08-23T10:31:51","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=246"},"modified":"2017-09-25T16:17:29","modified_gmt":"2017-09-25T16:17:29","slug":"graphing-trigonometric-functions-and-inverse-trigonometric-functions","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/graphing-trigonometric-functions-and-inverse-trigonometric-functions\/","title":{"rendered":"Graphing Trigonometric Functions and Inverse Trigonometric Functions"},"content":{"rendered":"
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\u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Graphing Trigonometric Functions and Inverse Trigonometric Functions<\/h1>\n

Objective<\/h4>\n

In this lesson, we will discuss how to prepare graphical representations of the trigonometric ratios tangent,\u00a0cotangent, secant, and cosecant. We will also define, discuss, and graph inverse trigonometric functions.<\/p>\n

Previously Covered:<\/h4>\n

The basic trigonometric functions are:<\/p>\n

    \n
  • <\/strong><\/li>\n
  • <\/li>\n
  • <\/strong><\/li>\n
  • <\/strong><\/li>\n
  • <\/strong><\/li>\n
  • A unit circle<\/em><\/strong> is an easy way to\u00a0illustrate the two most common angle measurements, degrees and
    \nradians, on a circle.<\/li>\n<\/ul>\n
    \n

    How do we graph trigonometric functions?<\/h3>\n

    You have been introduced to the six major trigonometric\u00a0functions (sine, cosines, tangent, cotangent, secant, and\u00a0cosecant) and their formulas, but it is important to be able to\u00a0recognize the graphs of these functions as well. So, in this\u00a0lesson, you will be introduced to the graphs of these functions.\u00a0Before we move ahead, we will first discuss odd and even\u00a0functions, since they can be recognized using these graphs.<\/p>\n

    An odd\u00a0function<\/abbr> is a function that always holds true for\u00a0,\u00a0and an even\u00a0function<\/abbr> is a function that always holds true for\u00a0.\u00a0Not all functions are either even or odd. Many (most, in fact) are\u00a0neither. When looking at a\u00a0graph, you can recognize an odd function from an even function by\u00a0knowing that an even function is symmetric around the <\/span>y<\/span><\/em>-axis.\u00a0This means that if you were to fold everything on the right side\u00a0of the <\/span>y<\/span><\/em>-axis\u00a0over the left side of the <\/span>y<\/span><\/em>-axis,\u00a0you would have the same left side of the <\/span>y<\/span><\/em>-axis\u00a0as you did before you folded over the right side onto the left\u00a0side.<\/span><\/p>\n

    What does the tangent function look like?<\/h4>\n

    Many mathematicians believe that the tangent is the most\u00a0important trigonometric function. We learned earlier in this\u00a0module that\u00a0.\u00a0One important feature of the tangent function is that it has\u00a0undefined function values at certain points. The sine and cosine\u00a0functions are defined on all real numbers, but the tangent\u00a0function is not.<\/p>\n

    We know that one way of getting an undefined point is by\u00a0dividing a number over zero. On the unit circle, we know that\u00a0\u00a0at 90\u00b0 and 270\u00b0, or\u00a0\u00a0and\u00a0\u00a0radians. We know that\u00a0,\u00a0so at\u00a0\u00a0there is an undefined point. If you move backward on the unit\u00a0circle, 270\u00b0 is at the same point as \u201390\u00b0, so there\u00a0is also an undefined point at \u20131.56. So, both\u00a0\u00a0and\u00a0\u00a0are the undefined points.<\/p>\n

    The tangent graph is not\u00a0symmetrical over the\u00a0<\/span>y<\/span><\/em>-axis, so the\u00a0tangent function is an odd function. Since all odd functions have\u00a0the form\u00a0,\u00a0we can say that\u00a0.
    \n<\/span><\/p>\n

    You are probably wondering\u00a0what is so special about the tangent function. A line that is\u00a0tangent to a function with respect to the <\/span>x<\/span><\/em>-axis\u00a0can also be referred to as the slope. You should remember the\u00a0definition and application of the slope from a previous module.\u00a0The fact that the tangent is equal to the slope is logically\u00a0reasonable because the tangent function is equal to the sine over\u00a0the cosine, which is the <\/span>y<\/span><\/em>-coordinate\u00a0over the <\/span>x<\/span><\/em>-coordinate,\u00a0and the slope, as you have learned, is equal to rise over run. <\/span><\/p>\n

    What does the graph of the cotangent function look like?<\/h3>\n

    Previously, we learned that the formula for the cotangent\u00a0function is<\/p>\n

    .<\/strong><\/p>\n

    Like the tangent function, the cotangent function has undefined\u00a0function values at certain points. Considering the unit\u00a0circle, you know that\u00a0,\u00a0and\u00a0,\u00a0so you have \u00a0<\/span>\u00a0<\/span><\/strong>which is\u00a0undefined. <\/span><\/p>\n

    The cotangent function is\u00a0an odd function because it is not symmetric about the <\/span>y<\/span><\/em>-axis.\u00a0So, since all odd functions have the form\u00a0,\u00a0we know that\u00a0.<\/span><\/p>\n

    What does the graph of the secant function look like?<\/h4>\n

    We know that the secant formula is<\/p>\n

    .<\/span><\/strong><\/p>\n

    The graph of this formula, like those of tangent and cotangent,\u00a0has undefined points. You should be able to figure out the exact\u00a0points that are undefined.<\/p>\n

    You know from the unit circle above that at 90\u00b0 and \u201390\u00b0\u00a0(\u00a0and\u00a0radians),\u00a0.\u00a0So the graph of the secant function shows undefined points at\u00a0\u00a0and at \u20131.57.<\/p>\n

    The graph is symmetric\u00a0about the <\/span>y<\/span><\/em>-axis,\u00a0so secant is an even function. Since all even functions have the\u00a0form\u00a0,\u00a0we know that\u00a0. <\/span><\/p>\n

    What does the graph of the cosecant function look like?<\/h4>\n

    The formula for cosecant is<\/p>\n

    .<\/strong><\/p>\n

    Since\u00a0\u00a0at 0\u00b0 and 180\u00b0, you can see that the cosecant function is\u00a0undefined at 0,\u00a0\u00a0(180\u00b0), and\u00a0.\u00a0It is undefined at\u00a0\u00a0because when moving backward on the unit circle, 180\u00b0 is at\u00a0the same point at \u2013180\u00b0. You should have noticed that on\u00a0these trigonometric functions, there may be (and most likely are)\u00a0other undefined points, but remember that radians can be\u00a0identified on a scale of infinity, and you can use the unit circle\u00a0to determine where the undefined points are on a scaled graph.<\/p>\n

    The cosecant is not symmetric about the y<\/em>-axis, so it\u00a0is an odd function. This means that\u00a0.<\/p>\n

    Now that you have learned how to identify odd functions and\u00a0even functions, you should understand why\u00a0\u00a0and\u00a0.<\/p>\n

    What are inverse trigonometric functions?<\/h3>\n

    You may be able to guess what inverse trigonometric functions\u00a0are. Inverse trigonometric functions can be written as\u00a0,\u00a0,\u00a0and\u00a0\u00a0or arcsin,\u00a0arccos,\u00a0and arctan.\u00a0The most important thing to remember when dealing with inverse\u00a0trigonometric functions is that\u00a0,\u00a0,\u00a0and\u00a0.\u00a0Cosecant is the reciprocal <\/em><\/strong>of sine,\u00a0while arcsin\u00a0is the inverse <\/em><\/strong>of sine. Inverse\u00a0trigonometric functions are all odd functions, so none of them are\u00a0symmetric about the\u00a0<\/span>y<\/span><\/em>-axis.<\/span><\/p>\n

    What is arcsin\u00a0\u00a0?<\/h4>\n

    The <\/span>inverse\u00a0sine function<\/span><\/abbr>(arcsin) is\u00a0just the inverse of the sine function. As mentioned above, it is\u00a0NOT the reciprocal of the sine function. This function can be\u00a0described the following way:\u00a0\u00a0when <\/span>y\u00a0<\/span><\/em>is between\u00a0\u00a0and\u00a0\u00a0such that\u00a0.<\/span><\/p>\n

    What is arccos\u00a0\u00a0?<\/h4>\n

    The inverse\u00a0cosine function<\/abbr> (arcos) is the inverse of the cosine function.\u00a0This function can be described in the following way:\u00a0\u00a0when y is between\u00a0\u00a0such that\u00a0.<\/p>\n

    What is the arctan,\u00a0and what does it look like?<\/h4>\n

    The <\/span>inverse\u00a0tangent function<\/span><\/abbr> (arctan) is shown below, and\u00a0\u00a0when <\/span>y\u00a0<\/span><\/em>is between\u00a0\u00a0and\u00a0,\u00a0such that\u00a0.
    \n<\/span><\/p>\n

    \"Inverse<\/p>\n

    Graphing Trigonometric Functions and Inverse Trigonometric Functions<\/h3>\n

    Now that we know about all the basic trigonometric functions\u00a0and inverse trigonometric functions, here are some standard points\u00a0associated with those angles. Compare the trigonometric functions\u00a0with the inverse trigonometric functions and try and to figure out\u00a0the patterns yourself. Memorize these common points or remember\u00a0the pattern, because they are useful and occur frequently.<\/p>\n

    If you have not figured out the pattern, take a look at the\u00a0sine and cosine trigonometric functions together. You may notice\u00a0that the pattern for sine is the square roots of\u00a0\u00a0and\u00a0\u00a0, and the pattern for cosine is the square roots of\u00a0\u00a0and\u00a0.
    \nThe pattern for tangent is simply the sine divided by the cosine.\u00a0Now, you should be able to determine the inverse trigonometric\u00a0functions using this information.<\/p>\n\n\n\n\n\n\n
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    Review of New Vocabulary and Concepts<\/h3>\n