Given that cos t = 5/13 and 0 is less than or equal to t less than or equal to pi over 2 find the exact value of sin t and tan 2t

Answer 1

cost = 5/13 = adjacent side over hypotenuse = a/h

sint = opposite side over hypotenuse = o/h

h^2 = o^2 + a^2; o^2 = h^2 - a^2, o = sqr(h^2- a^2) = sqr(13^2 -5^2) = sqr(169-25) = sqr144 = 12

sint = 12/13

t = about 67.4 in quadrant I, 0<t<pi/2

tan t = 2.4

2t = 134.8 in quadrant II

tan134.8 = about -1.008

tan135 = exactly -1

sin2t = 2sintcost = 2(12/13)(5/13 = 120/169

cos(2t) = cos^2(t )- sin^2(t) = (5/13)^ - (12/13)^2 = 25/169-144/169 = -119/169

tan(2t) = sin(2t)/cos(2t) = -120/119 = about -1.008

tan(2t) = exactly -120/119

sin(t) =exactly 120/169