# Sin 1 sin 7 π 5 18 5 pts write sincos 1 u sin 1 v as

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sin -1 (sin(-7 π /5)) = 18. (5 pts.) Write sin(cos -1 (u ) + sin -1 (v)) as an algebraic expression containing u and v. sin(cos -1 (u ) + sin -1 (v)) = 19. (5 pts.) Find the exact value of tan(2 tan -1 (3/4)). tan(2 tan -1 (3/4)) = 20. (5 pts.) Carefully sketch the graph of y = cos -1 (x). Label very carefully. y x 21. (5 pts.) Carefully sketch the graph of y = tan -1 (x). Label very carefully. y x

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Final Exam/MAC1114 Page 6 of 8 22. (10 pts.) Establish the following identity. 1 - cos( α ) sin( α ) = sin( α ) 1 + cos( α ) 23. (15 pts.) Very carefully complete the following derivation of a couple of half angle identities by giving the information requested and performing the computations needed. (a) Write down the trigonometric identity giving cos(x + y) in terms of sums of products of sines and cosines of x’s and y’s. cos(x + y) = (b) Uniformly replace each instance of a y by x in the identity above to obtain an identity for cos(2x). Use exponentiation to clean up the products appearing. cos(2x) = cos(x + x) = (c) Using the Pythagorean identity connecting sin 2 (x) and cos 2 (x), replace sin 2 (x) in the identity for cos(2x) and then solve for cos 2 (x) in terms of what remains. cos 2 (x) = (d) Uniformly replace x by θ /2 in the identity for cos 2 (x) that you obtained in part (c) above in order to obtain an identity for cos 2 ( θ /2). Clean up the algebra. cos 2 ( θ /2) = (e) Modify steps (c) and (d) appropriately to obtain a corresponding identity for sin 2 ( θ /2). Show your work neatly.
• Spring '08
• Storfer
• Trigonometry, Tan, pts, Final Exam/MAC1114

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