SOLUTION: Find all exact solutions on [0, 2𝜋). (Enter your answers as a comma-separated list.) 2 cos^2(t) + cos(t) = 1

Algebra ->  Trigonometry-basics -> SOLUTION: Find all exact solutions on [0, 2𝜋). (Enter your answers as a comma-separated list.) 2 cos^2(t) + cos(t) = 1      Log On


   

Question 1205306: Find all exact solutions on [0, 2𝜋). (Enter your answers as a comma-separated list.)
2 cos^2(t) + cos(t) = 1
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

given:
2cos%5E2%28t%29+%2B+cos%28t%29+=+1
Find all exact solutions on [0, 2pi)

2cos%5E2%28t%29+%2B+cos%28t%29+=+1
2cos%5E2%28t%29+%2B+cos%28t%29+-+1=0...write cos%28t%29+ as -cos%28t%29+%2B2cos%28t%29+
2cos%5E2%28t%29+-cos%28t%29%2B2cos%28t%29+-+1=0...group
%282cos%5E2%28t%29+%2B2cos%28t%29%29-%28cos%28t%29+%2B1%29=0
2cos%28t%29%28cos%28t%29+%2B1%29-%28cos%28t%29+%2B1%29=0
%28cos%28t%29+%2B+1%29+%282cos%28t%29+-+1%29+=+0
solutions:
%28cos%28t%29+%2B+1%29+=0 =>cos%28t%29+=-1+=>t=cos%5E-1%28-1%29 => t=pi
%282cos%28t%29+-+1%29+=+0+=> cos%28t%29+=1%2F2 =>t=cos%5E-1%281%2F2%29 => t=pi%2F3
all exact solutions on [0, 2pi)
t=pi
t=pi%2F3
t=5pi%2F3


Answer by ikleyn(52824) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find all exact solutions on [0, 2𝜋). (Enter your answers as a comma-separated list.)
2 cos^2(t) + cos(t) = 1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Write the original equation in the standard quadratic form 

    2cos%5E2%28t%29+%2B+cos%28t%29+-+1 = 0.


Factor left side

    (2cos(t)-1) * (cos(t)+1) = 0.


Case 1.  2cos(t) - 1 = 0  --->  cos(t) = 1%2F2  --->  t = pi%2F3%29  or  t = 5pi%2F3.


Case 2.   cos(t) + 1 = 0  --->  cos(t) = -1  --->  t = pi.


ANSWER.  The set of solutions is  pi%2F3,  pi  and  5pi%2F3,  in ascending order.

Solved.