Expert: Socrates - 2/21/2008

Question
Solve the equation for theta:
sec(theta + 10degrees)= csc(2theta + 20degrees)

Any help is greatly appreciated!

Answer
I will use "t" for the angle theta

sec(t+10) = 1/cos(t+10)

csc(2t+20) = 1/sin(2t+20)

Use the double angle formula

sin(2t+20) = 2sin(t+10)cos(t+10)


so

1/cos(t+10) = 1/2sin(t+10)cos(t+10)


multiply both sides by cos(t+10)


1 = 1/2sin(t+10)


multiply both sides by sin(t+10)


sin(t+10) = 1/2


so t+10 = 30

and t = 20 degrees

t = 20 degrees is the only acute angle solution


Since we want sin(t+10) = 1/2
t+10 could also be 150 degrees , so this gives t = 140 degres for another solution.

The answer is theta = 20 degrees or theta = 140 degrees , both angles will work.