{"id":1676,"date":"2023-08-31T17:06:09","date_gmt":"2023-08-31T17:06:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/?post_type=chapter&p=1676"},"modified":"2023-09-20T14:18:24","modified_gmt":"2023-09-20T14:18:24","slug":"inverse-trig-graphs-and-application","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/inverse-trig-graphs-and-application\/","title":{"raw":"Inverse Trig Graphs and Application","rendered":"Inverse Trig Graphs and Application"},"content":{"raw":"

Inverse Trigonometric Functions<\/h3>\r\n

The six basic trigonometric functions are periodic, and therefore they are not one-to-one. However, if we restrict the domain of a trigonometric function to an interval where it is one-to-one, we can define its inverse. Consider the sine function (Figure 1.34<\/a>). The sine function is one-to-one on an infinite number of intervals, but the standard convention is to restrict the domain to the interval [<\/span>\u2212<\/span>\u03c0\/<\/span>2<\/span><\/span>,<\/span>\u03c0\/<\/span>2<\/span><\/span><\/span>]<\/span><\/span>.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> By doing so, we define the inverse sine function on the domain [<\/span>\u22121<\/span>,<\/span>1<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> such that for any x<\/span><\/span><\/span><\/span><\/span><\/span><\/span> in the interval [<\/span>\u22121<\/span>,<\/span>1<\/span>]<\/span>,<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> the inverse sine function tells us which angle \u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span> in the interval [<\/span>\u2212<\/span>\u03c0\/<\/span>2<\/span><\/span>,<\/span>\u03c0\/<\/span>2<\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> satisfies sin<\/span><\/span>\u03b8<\/span>=<\/span>x<\/span>.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> Similarly, we can restrict the domains of the other trigonometric functions to define inverse trigonometric functions<\/span>, which are functions that tell us which angle in a certain interval has a specified trigonometric value.<\/p>\r\n\"\"\"\"\r\n\r\n \r\n

\r\n\r\n[ohm_question sameseed=1 hide_question_numbers=1]15113[\/ohm_question]\r\n\r\n<\/div>\r\n
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Exercises<\/h3>\r\n\\[ohm_question sameseed=1 hide_question_numbers=1]61751[\/ohm_question]\r\n\r\n<\/div>\r\n
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Try It<\/h3>\r\n[ohm_question sameseed=1 hide_question_numbers=1]66845[\/ohm_question]\r\n\r\n<\/div>\r\n
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Try It<\/h3>\r\n[ohm_question sameseed=1 hide_question_numbers=1]66842[\/ohm_question]\r\n\r\n<\/div>\r\n
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Examples<\/h3>\r\n[ohm_question sameseed=1 hide_question_numbers=1]66846[\/ohm_question]\r\n\r\n<\/div>\r\n
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Try It<\/h3>\r\n[ohm_question sameseed=1 hide_question_numbers=1]138872[\/ohm_question]\r\n\r\n<\/div>\r\n
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Examples<\/h3>\r\n[ohm_question sameseed=1 hide_question_numbers=1]4448[\/ohm_question]\r\n\r\n<\/div>\r\n ","rendered":"

Inverse Trigonometric Functions<\/h3>\n

The six basic trigonometric functions are periodic, and therefore they are not one-to-one. However, if we restrict the domain of a trigonometric function to an interval where it is one-to-one, we can define its inverse. Consider the sine function (Figure 1.34<\/a>). The sine function is one-to-one on an infinite number of intervals, but the standard convention is to restrict the domain to the interval [<\/span>\u2212<\/span>\u03c0\/<\/span>2<\/span><\/span>,<\/span>\u03c0\/<\/span>2<\/span><\/span><\/span>]<\/span><\/span>.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> By doing so, we define the inverse sine function on the domain [<\/span>\u22121<\/span>,<\/span>1<\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> such that for any x<\/span><\/span><\/span><\/span><\/span><\/span><\/span> in the interval [<\/span>\u22121<\/span>,<\/span>1<\/span>]<\/span>,<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> the inverse sine function tells us which angle \u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span> in the interval [<\/span>\u2212<\/span>\u03c0\/<\/span>2<\/span><\/span>,<\/span>\u03c0\/<\/span>2<\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> satisfies sin<\/span><\/span>\u03b8<\/span>=<\/span>x<\/span>.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> Similarly, we can restrict the domains of the other trigonometric functions to define inverse trigonometric functions<\/span>, which are functions that tell us which angle in a certain interval has a specified trigonometric value.<\/p>\n

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