HW12sol - MAC1114 - Homework 12 Due Thursday - April 7 1....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAC1114 - Homework 12 Due Thursday - April 7 1. Recall that cos(u + v ) = cos u cos v − sin u sin v . (a) Show that cos(2u) = 1 − 2 sin2 u. cos(2u) = cos(u + u) = cos u cos u − sin u sin u = cos2 u − sin2 u = (1 − sin2 u) − sin2 u = 1 − 2 sin2 u 1 − cos 2u (b) Use part (a) to show that sin2 u = 2 cos(2u) = 1 − 2 sin2 u ⇒ 2 sin2 u = 1 − cos(2u) ⇒ sin2 u = 1−cos(2u) 2 (c) Use part (b) to show that, for v in quadrant I, sin v = 2 1 − cos v 2 Take the identity in part (b) and substitute u = v to get sin2 v = 1−cos v ⇒ sin v = ± 1−cos v . 2 2 2 2 2 Since v is in Quadrant I, we use the positive part (actually this is technically incorrect, I should π π have said 0 ≤ v ≤ π . The angle v = 94 is in quadrant I for example, but v = 98 is in quadrant 2 2 v III, so sin 2 < 0 and we'd want the negative piece). 2. Solve the equation tan x − sin x = 0 in the interval [0, 2π ). 2 See your class notes. 3. If cos x = 1 5 and x is in Quadrant I, nd sin 4x. Draw a triangle or use the identity sin2 x +cos2 x = 1 to nd that sin x = 2 2 sin(2 · 2x) = 2 sin(2x) cos(2x) = 2(2 sin x cos x)(cos x − sin x) = 2(2 1 √ 24 5 . Now note that sin(4x) √ 24 1 1 )( 25 − 24 ) = 4 24(−23) . 55 25 625 √ = ...
View Full Document

This note was uploaded on 06/10/2011 for the course MAC 1114 taught by Professor Gentimis during the Spring '11 term at University of Florida.

Ask a homework question - tutors are online