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Math 140 Since tan 6 sin 6 / cos 6 12 / Product-to-sum rules RECALL cosx y cos x cos y sin x sin y cosx y cos x cos y sin x sin y Adding these gives cosx y cosx y 2 cos x cos y Subtracting gives cosx y cosx y 2 sin x sin y Similarly sinx y sin x cos y cos x sin y sinx y sin x cos y cos x sin y Adding these gives sinx y sinx y 2 sin x cos y RECALL. sin(x) and cos(x) are always between -1 and 1. `cos x 3 iff never. tno solution. `2 cos 2 x cos x 1 2 cos 2 x cos x 1 0 2 cos x 1cos x 1 0 cos x 12 or cos x 1 x = p/3 + 2pn or x = -p/3 + 2pn or x = p + 2pn. Dividing by 2 gives PRODUCT-TO-SUM FORMULA. sin x sin y 12 [cosx y cosx y ] cos x cos y 12 [cosx y cosx y ] sin x cos y 12 [sinx y sinx y ] Write as a sum or difference of trigonometric functions. `sin 2 cos 6 12 sin 2 6 sin 2 6 12 sin 3 sin 2 3 . You could continue and get: 3 /2. `cos x cos 2x 12 cosx 2x cosx 2x 12 cosx cos3x 12 cosx cos3x. Solving trigonometric equations RECALL. sin x sin x, cos x cosx. Note: x and x are called supplementary angles. Find all solutions for each equation. `sin 12 y=1/2 p-p/6=5p/6 3 2 13 p/6 is one solution. Since tan(x) has period p, adding a multiple of p also gives a solution. The general solution is x = p/6 + pn. For sin, cos, add 2pn to the (usually two) simplest solutions. For tan, add pn to the one simplest solution. `sin 2x 1 iff 2x = p/2 + 2pn, Only one simple solution here. iff x = p/4 + pn. Lecture 23 p/6 `sin 2 x cos x 1 0 1 cos 2 x cos x 1 0 cos 2 x cos x 2 0 cos 2 x cos x 2 0 cos x 1cos x 2 cos x 1 or cos x 2The second is impossible. x = p + 2pn. `2 tan 2 x 3 tan x sec x 2 sec 2 x 0 2 sin 2 x 3 sin x 2 0 2 sin x 1sin x 2 0 sin x 12 or sin x 2 The second is impossible. x = -p/6 + 2pn or x = -5p/6 + 2pn. RECALL. A function is 1-1 iff no horizontal line crosses its graph more than once. sin, cos and tan are not 1-1. But they are 1-1 on the heavily marked intervals. These are the largest such intervals containing first quadrant angles. sin -p/2 The two simplest solutions: q = p/6, and q = p-p/6 = 5p/6. Since sin(q) has period 2p, adding a multiple of 2p to either of these also produces a solution. The general solution consists of the two sets q = p/6 + 2pn or q = 5p/6 + 2pn for n an integer. `cos x 12 Here we use x for the angle instead of q. The two simplest solutions are x = p/3 and x = -p/3. The general solution is x = p/3 + 2pn or x = -p/3 + 2pn for n an integer. CONVENTION. Assume n is an arbitrary, possibly negative, integer. We’ll omit the phrase “for n an integer”. `tan x 13 p/2 cos p 0 tan -p/2 p/2 For sin, the interval is [-p/2, p/2]. For cos, the interval is [0, p]. For tan, the interval is (-p/2, p/2).