Sin(arccos(-9/41 )+ 2arctan(-60/11)) I need help solving showing all work and st
ID: 3192145 • Letter: S
Question
Sin(arccos(-9/41 )+ 2arctan(-60/11)) I need help solving showing all work and steps. Answer must be exact, no calculator allowed. I know the sum identities but the 2arctan is throwing me off. This can get really confusing fast so a new line for each step is appreciated.Explanation / Answer
sin(a + b) => sin(a)cos(b) + sin(b)cos(a) sin(2a) = 2sin(a)cos(a) cos(2a) = 2cos(a)^2 - 1 sin(t)^2 + cos(t)^2 = 1 csc(t)^2 - cot(t)^2 = 1 sec(t)^2 - tan(t)^2 = 1 sin(arccos(-9/41) + 2 * arctan(-60/11)) => sin(arccos(-9/41)) * cos(2 * arctan(-60/11)) + cos(arccos(-9/41)) * sin(2 * arctan(-60/11)) => sin(arccos(-9/41)) * (2cos(arctan(-60/11))^2 - 1) + cos(arccos(-9/41)) * 2 * sin(arctan(-60/11)) * cos(arctan(-60/11)) => sin(arccos(-9/41)) * (2/sec(arctan(-60/11))^2 - 1) + 2 * (-9/41) * sqrt(1 - cos(arctan(-60/11))^2) * 1/sec(arctan(-60/11)) => Okay, confused yet? Let's look at each part: sin(arccos(-9/41)) => sqrt(sin(arccos(-9/41))^2) => sqrt(1 - cos(arccos(-9/41))^2) => sqrt(1 - (-9/41)^2) => sqrt(1 - 81/1681) => sqrt((1681 - 81) / 1681) => sqrt(1600 / 1681) => +/- 40/41 (2/sec(arctan(-60/11))^2 - 1) => 2 / (1 + tan(arctan(-60/11))^2) - 1 => 2 / (1 + (-60/11)^2) - 1 => 2 / (1 + 3600/121) - 1 => 2 / (3721 / 121) - 1 => 242 / 3721 - 1 => (242 - 3721) / 3721 => -3479/3721 sqrt(1 - cos(arctan(-60/11))^2) sqrt(1 - 1/sec(arctan(-60/11))^2) sqrt(1 - 1/(1 + tan(arctan(-60/11))^2) sqrt(1 - 1/(1 + 3600/121)) sqrt(1 - 1/(3721/121)) sqrt(1 - 121/3721) sqrt((3721 - 121) / 3721) sqrt(3600/3721) +/- 60/61 1/sec(arctan(-60/11)) 1/sqrt(sec(arctan(-60/11))^2) 1/sqrt(1 + tan(arctan(-60/11))^2) 1/sqrt(1 + 3600/121) 1/sqrt(3721/121) sqrt(121/3721) +/- 11/61 sin(arccos(-9/41)) * (2/sec(arctan(-60/11))^2 - 1) + 2 * (-9/41) * sqrt(1 - cos(arctan(-60/11))^2) * 1/sec(arctan(-60/11)) => (+/- 40/41) * (-3479 / 3761) - (18/41) * (+/- 60/61) * (+/- 11/61) => +/- (1/41) * (1/3761) * (40 * 3479) +/- (1/41) * (1/3761) * (18 * 60 * 11) => +/- (1/41) * (1/3761) * 20 * (2 * 3479 +/- 3 * 18 * 11) +/- (1/41) * (1/3761) * 40 * (3479 +/- 297) +/- (40/154201) * (3479 +/- 297) +/- (40/154201) * 3776 , +/- (40/154201) * 3182 +/- 127280/154201 , +/- 151040/154201