How do you use #csctheta=5# to find #cottheta#?

2 Answers
Mar 27, 2017

#cottheta=+-2sqrt6#

Explanation:

The terminal ray of an angle #theta# in standard position intersects a point #(x,y)# on the unit circle such that #csctheta=1/y#, #cottheta=x/y#, and #x^2+y^2=1#.

Knowing that #csctheta=5# we know that #y=1/5#

Since #x^2+y^2=1#, #x=+-sqrt(1-y^2)#

#cottheta=x/y=(+-sqrt(1-y^2))/y=(+-sqrt(1-(1/5)^2))/(1/5)#
#cottheta=+-5sqrt(1-1/25)=+-5sqrt(24/25)=+-cancel5sqrt24/cancel5=+-2sqrt6#

Mar 27, 2017

#+- 2sqrt6#

Explanation:

Use trig identity:
#1 + cot^2 t = csc^2 t#
In this case:
#1 + cot^2 t = 25#
#cot^2 t = 24#
#cot t = +- 2sqrt6#