Exact Value of sin 18 Project: You have learned how to find exact values of functions of

Chapter 5, Problem C.1

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Exact Value of sin 18 Project: You have learned how to find exact values of functions of multiples of 30 and 45. By the composite argument property, you found an exact value for sin 15 by writing it as sin (45 30). In this problem you will combine trigonometric properties with algebraic techniques and some ingenuity to find an exact value of sin 18. a. Use the double argument property for sine to write an equation expressing sin 72 in terms of sin 36 and cos 36. b. Transform the equation in part a so that sin 72 is expressed in terms of sin 18 and cos 18. You should find that the sine form of the double argument property for cos 36 works best. c. Recall by the cofunction property that sin 72 = cos 18. Replace sin 72 in your equation from part b with cos 18. If you have done everything correctly, the cos 18 should disappear from the equation, leaving a cubic (third-degree) equation in sin 18. may help to let x = sin 18 and solve for x. If you rearrange the equation so that the right side is 0, you should find that (2x 1) is a factor of the left side. You can find the other factor by long division or synthetic substitution. To find the exact solutions, recall the multiplication property of zero and the quadratic formula. equation in part d. Only one of these solutions is possible. Which solution? of sin : See if you can extend this pattern to sines of other angles! C