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4-6 Inverse Trigonometric Functions Find the exact value of each expression, if it exists. 1. sin – 1 0 SOLUTION: with a y-coordinate of 0. Find a point on the unit circle on the interval When t = 0, sin t = 0. Therefore, sin –1 0 = 0. 3. arcsin SOLUTION: with a y-coordinate of Find a point on the unit circle on the interval When t = , sin t = . Therefore, arcsin = . . 5. SOLUTION: with a y-coordinate of Find a point on the unit circle on the interval eSolutions Manual - Powered by Cognero When t = , sin t = . Therefore, sin –1 . Page 1 = . 4-6 When Inverse Functions t = Trigonometric , sin t = . Therefore, arcsin = . 5. SOLUTION: with a y-coordinate of Find a point on the unit circle on the interval When t = , sin t = . Therefore, sin –1 = . . 7. cos– 1 SOLUTION: Find a point on the unit circle on the interval When t = , cos t = –1 . Therefore, cos with an x-coordinate of = . . 9. SOLUTION: Find a point on the unit circle on the interval eSolutions Manual - Powered by Cognero with an x-coordinate of . Page 2 –1 4-6 When Inverse Functions t = Trigonometric , cos t = . Therefore, cos = . 9. SOLUTION: Find a point on the unit circle on the interval When t = , cos t = . Therefore, with an x-coordinate of = . . 11. arctan 1 SOLUTION: Find a point on the unit circle on the interval When t = , tan t = . Therefore, arctan 1= such that =1. . 13. SOLUTION: Find a point on the unit circle on the interval eSolutions Manual - Powered by Cognero such that = . Page 3 When t = , tan t = . Therefore, arctan 1= . 4-6 Inverse Trigonometric Functions 13. SOLUTION: such that Find a point on the unit circle on the interval When t = , tan t = . Therefore, tan –1 = = . . 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. with a y-coordinate of Find a point on the unit circle on the interval When t = , sin t = . Therefore, sin –1 = . . Sketch the graph of each function. 17. y = arcsin x SOLUTION: eSolutions Manual - Powered by Cognero First, rewrite y = arcsin x in the form sin y = x. Page 4 –1 4-6 When Inverse Functions t = Trigonometric , sin t = . Therefore, sin = . Sketch the graph of each function. 17. y = arcsin x SOLUTION: First, rewrite y = arcsin x in the form sin y = x. Next, assign values to y on the interval to make a table of values. y x = sin y –1 0 0 1 Plot the points and connect them with a smooth curve. 19. y = sin – 1 (x + 3) SOLUTION: First, rewrite y = sin –1 (x + 3) in the form sin y = x. Next, assign values to y on the interval to make a table of values. y x = sin y –3 –3.89 0 –3.85 –3 eSolutions Manual - Powered by Cognero –2.15 –2.11 Page 5 4-6 Inverse Trigonometric Functions 19. y = sin – 1 (x + 3) SOLUTION: First, rewrite y = sin –1 (x + 3) in the form sin y = x. to make a table of values. Next, assign values to y on the interval y x = sin y –3 –3.89 0 –3.85 –3 –2.15 –2.11 Plot the points and connect them with a smooth curve. 21. y = arccos x SOLUTION: First, rewrite y = arccos x in the form cos y = x. Next, assign values to y on the interval x = cos y y 0 to make a table of values. 1 eSolutions Manual - Powered by Cognero 0 Page 6 4-6 Inverse Trigonometric Functions 21. y = arccos x SOLUTION: First, rewrite y = arccos x in the form cos y = x. Next, assign values to y on the interval x = cos y y 0 to make a table of values. 1 0 -1 Plot the points and connect them with a smooth curve. 23. y = arctan x SOLUTION: First, rewrite y = arctan x in the form tan y = x. Next, assign values to y on the interval to make a table of values. y x = tan y –3.08 0 –1 0 1 3.08 Plot the points and connect them with a smooth curve. eSolutions Manual - Powered by Cognero Page 7 4-6 Inverse Trigonometric Functions 23. y = arctan x SOLUTION: First, rewrite y = arctan x in the form tan y = x. Next, assign values to y on the interval to make a table of values. y x = tan y –3.08 –1 0 0 1 3.08 Plot the points and connect them with a smooth curve. 25. y = tan– 1 (x + 1) SOLUTION: –1 First, rewrite y = tan (x + 1) in the form tan y = x. Next, assign values to y on the interval for y-values of to make a table of values. Note that x = tan y − 1 has no x-values and − . y x = tan y – 1 0 –2 –1 0 eSolutions Manual - Powered by Cognero Plot the points and connect them with a smooth curve. Page 8 4-6 Inverse Trigonometric Functions 25. y = tan– 1 (x + 1) SOLUTION: –1 First, rewrite y = tan (x + 1) in the form tan y = x. Next, assign values to y on the interval for y-values of to make a table of values. Note that x = tan y − 1 has no x-values and − . y x = tan y – 1 0 –2 –1 0 Plot the points and connect them with a smooth curve. Further investigation reveals that as x approaches negative infinity, y approaches , and as x approaches positive infinity, y approaches . eSolutions Manual - Powered by Cognero Page 9