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When t = 4-6 Inverse Trigonometric Functions 1. sin = . Therefore, arcsin . Find the exact value of each expression, if it exists. –1 , sin t = 3. arcsin 0 SOLUTION: SOLUTION: Find a point on the unit circle on the interval Find a point on the unit circle on the interval with a y-coordinate of with a y-coordinate of 0. When t = 0, sin t = 0. Therefore, sin –1 When t = 0 = 0. , sin t = . . Therefore, arcsin = . 2. arcsin 4. sin – 1 SOLUTION: Find a point on the unit circle on the interval SOLUTION: with a y-coordinate of . Find a point on the unit circle on the interval with a y-coordinate of . When t = , sin t = . Therefore, arcsin = When t = . , sin t = . Therefore, sin –1 = . 5. 3. arcsin SOLUTION: SOLUTION: Find a point on the unit circle on the interval Find a point on the unit circle on the interval with a y-coordinate of eSolutions Manual - Powered by Cognero . with a y-coordinate of . Page 1 –1 4-6 When Inverse Functions t = Trigonometric , sin t = . Therefore, sin = When t = . , cos t = 0. Therefore, arccos 0 = . 7. cos– 1 5. SOLUTION: SOLUTION: Find a point on the unit circle on the interval Find a point on the unit circle on the interval with a y-coordinate of When t = = , sin t = with an x-coordinate of . . Therefore, sin –1 When t = , cos t = . –1 . . 8. arccos (–1) 6. arccos 0 SOLUTION: SOLUTION: Find a point on the unit circle on the interval with an x-coordinate of 0. Find a point on the unit circle on the interval with an x-coordinate of –1. When t = = . Therefore, cos , cos t = 0. Therefore, arccos 0 = When t = . , cos t = –1. Therefore, arccos (–1)= . 9. –1 7. cos SOLUTION: Find a point on the unit circle on the interval SOLUTION: Find a point on the unit circle on the interval with an x-coordinate of eSolutions Manual - Powered by Cognero with an x-coordinate of . . Page 2 4-6 Inverse Trigonometric Functions When t = , cos t = –1. Therefore, arccos (–1)= . When t = , cos t = –1 . Therefore, cos = . 15. ARCHITECTURE The support for a roof is 9. shaped like two right triangles, as shown below. Find θ. SOLUTION: Find a point on the unit circle on the interval with an x-coordinate of . SOLUTION: Use inverse trigonometric functions and the unit circle to solve. Find a point on the unit circle on the interval with a y-coordinate of When t = , cos t = . Therefore, . = . 10. cos– 1 SOLUTION: Find a point on the unit circle on the interval with an x-coordinate of When t = , sin t = . Therefore, sin –1 = . . 16. RESCUE A cruise ship sailed due west 24 miles before turning south. When the cruise ship became disabled and the crew radioed for help, the rescue boat found that the fastest route covered a distance of 48 miles. Find the angle θ at which the rescue boat should travel to aid the cruise ship. When t = , cos t = –1 . Therefore, cos = . 15. ARCHITECTURE The support for a roof is shaped like two right triangles, as shown below. Find θ. SOLUTION: Use inverse trigonometric functions and the unit circle to solve. Find a point on the unit circle on the interval with an x-coordinate of SOLUTION: eSolutions Manual - Powered by Cognero Use inverse trigonometric functions and the unit circle to solve. . Page 3 When t = –1 4-6 When Inverse Functions t = Trigonometric , sin t = . Therefore, sin = . 16. RESCUE A cruise ship sailed due west 24 miles before turning south. When the cruise ship became disabled and the crew radioed for help, the rescue boat found that the fastest route covered a distance of 48 miles. Find the angle θ at which the rescue boat should travel to aid the cruise ship. , cos t = –1 . Therefore, cos = . Sketch the graph of each function. 17. y = arcsin x SOLUTION: First, rewrite y = arcsin x in the form sin y = x. to Next, assign values to y on the interval make a table of values. y x = sin y –1 SOLUTION: Use inverse trigonometric functions and the unit circle to solve. Find a point on the unit circle on the interval with an x-coordinate of 0 0 1 . Plot the points and connect them with a smooth curve. 21. y = arccos x When t = , cos t = –1 . Therefore, cos = SOLUTION: First, rewrite y = arccos x in the form cos y = x. . Sketch the graph of each function. 17. y = arcsin x Next, assign values to y on the interval make a table of values. x = cos y y SOLUTION: First, rewrite y = arcsin x in the form sin y = x. 0 to 1 Next, assign values to y on the interval to 0 make a table of values. eSolutions Manual - Powered by Cognero y -1 x = sin y –1 Plot the points and connect them with a smooth Page 4 lies on the interval [–1, 1]. Therefore, = 4-6 Inverse Trigonometric Functions The inverse property applies, because 21. y = arccos x . 30. SOLUTION: First, rewrite y = arccos x in the form cos y = x. SOLUTION: The inverse property applies, because lies on the interval [–1, 1]. Therefore, = to Next, assign values to y on the interval make a table of values. x = cos y y 0 . 1 0 -1 Plot the points and connect them with a smooth curve. Find the exact value of each expression, if it exists. 29. SOLUTION: The inverse property applies, because lies on the interval [–1, 1]. Therefore, = . 30. SOLUTION: TheManual inverse property eSolutions - Powered by applies, Cognero interval [–1, 1]. Therefore, because lies on the = . Page 5