SOLUTION: Suppose cos(t)=-1/3 and sin(t)<0 Use appropriate identities to find each of the following? sec(-t) sin(-t) cos(t+5&#960;/3) sin(2t) I can find sin sin^2t = 1-c

Algebra ->  Trigonometry-basics -> SOLUTION: Suppose cos(t)=-1/3 and sin(t)<0 Use appropriate identities to find each of the following? sec(-t) sin(-t) cos(t+5&#960;/3) sin(2t) I can find sin sin^2t = 1-c      Log On


   

Question 925282: Suppose cos(t)=-1/3 and sin(t)<0 Use appropriate identities to find each of the following?
sec(-t)
sin(-t)
cos(t+5π/3)
sin(2t)

I can find sin
sin^2t = 1-cos^2t
sin^2t = 1-(1/9)
sin^2t=8/9
sin^2t= -(2√2)/3


Please give details I have to understand
Thanks!
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
You are right in that
sin%5E2%28t%29=8%2F9 and
sin%28t%29=+-2sqrt%282%29%2F3 (since sin%28t%29%3C0 as the problem says)

For any t , sin%28-t%29=-sin%28t%29 , so
highlight%28sin%28-t%29=2sqrt%282%29%2F3%29

Using trigonometric identity formulas,
you can find cos%28t%2B5pi%2F3%29 and sin%282t%29 :

One of the most popular trigonometric identities is
cos%28A%2BB%29=cos%28A%29%2Acos%28B%29-sin%28A%29%2Asin%28B%29 , so
cos%28t%2B5pi%2F3%29=cos%28t%29%2Acos%285pi%2F3%29-sin%28t%29%2Asin%285pi%2F3%29 ,
and since cos%285pi%2F3%29=cos%285pi%2F3-2pi%29=cos%28-pi%2F3%29=cos%28pi%2F3%29=1%2F2
and sin%285pi%2F3%29=sin%285pi%2F3-2pi%29=sin%28-pi%2F3%29=-sin%28pi%2F3%29=-sqrt%283%29%2F2

cos%28t%2B5pi%2F3%29+=-1%2F6-%282sqrt%282%29sqrt%283%29%2F%282%2A3%29%29
highlight%28cos%28t%2B5pi%2F3%29+=-1%2F6-sqrt%286%29%2F3%29

An other popular trigonometric identity is
sin%282A%29=2sin%28A%29cos%28A%29 , so
sin%282t%29=2sin%28t%29cos%28t%29
sin%282t%29=2%28-2sqrt%282%29%2F3%29%28-1%2F3%29
highlight%28sin%282t%29=4sqrt%282%29%2F9%29