First find cos t by using trig identity: cos^2 t = 1 - sin^2 tcos2t=1−sin2t
cos^2 t = 1 - 9/25 = 16/25cos2t=1−925=1625 --> cos t = +- 4/5cost=±45.
Since t is in Quadrant IV, then cos t is positive --> cos t = 4/5
Next find sin (t/2) and cos (t/2) by using trig identities:
1 + cos 2t = 2cos^2 t1+cos2t=2cos2t
1 - cos 2t = 2sin^2 t1−cos2t=2sin2t
Find cos (t/2)cos(t2)
2cos^2 (t/2) = 1 + cos t = 1 + 4/5 = 9/52cos2(t2)=1+cost=1+45=95
cos^2 (t/2) = 9/10cos2(t2)=910
cos (t/2) = 3/sqrt10 = 3sqrt10/10cos(t2)=3√10=3√1010 (because cos (t/2) is positive).
Find sin (t/2)
2sin^2 (t/2) = 1 - cos t = 1 - 4/5 = 1/52sin2(t2)=1−cost=1−45=15
sin^2 (t/2) = 1/10sin2(t2)=110
sin (t/2) = +- 1/sqrt10 = +- sqrt10/10sin(t2)=±1√10=±√1010
Since t is in Quadrant IV, sin t is negative
tan (t/2) = sin/(cos) = (- sqrt10/10)(10/(3sqrt10)) = - 1/3 tan(t2)=sincos=(−√1010)(103√10)=−13
