SOLUTION: Solve the equation (2 cos θ + 1)(tan θ − 1) = 0 for 0 ≤ θ ≤ 2π

Algebra ->  Trigonometry-basics -> SOLUTION: Solve the equation (2 cos θ + 1)(tan θ − 1) = 0 for 0 ≤ θ ≤ 2π      Log On


   

Question 1071165: Solve the equation (2 cos θ + 1)(tan θ − 1) = 0 for 0 ≤ θ ≤ 2π
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
The equation is true if either of these is true:
+2%2Acos%28+theta+%29+%2B+1+=+0+
+2%2Acos%28+theta+%29+=+-1+
(1) +cos%28+theta+%29+=+-1%2F2+
OR
+tan%28+theta+%29+-+1+=+0+
(2) +tan%28+theta+%29+=+1+
--------------------------
(1) +theta+=+arc+cos%28+-1%2F2+%29+
(1) +theta+=+2.0944+
If I divide this by +pi+ I get +2%2F3+, which tells me
(1) +theta++=+%28+2%2Api+%29%2F3+
This is +theta+=+pi+-+pi%2F3+
The cosine is also negative for
+theta+=+pi+%2B+pi%2F3+ which is
+theta+=+%28+4%2Api+%29%2F3+
--------------------------
(2) +theta+=+arc+tan%28+1+%29+
(2) +theta+=+.7854+
Dividing by +pi+, I get +1%2F4+, so
(2) +theta+=+pi%2F4+
The tan function is also positive in the 3rd quadrant, so
(2) +theta+=+pi+%2B+pi%2F4+
(2) +theta+=+%28+5%2Api+%29%2F4+
-----------------------------
The solutions are:
+theta+ = +%28+2%2Api+%29%2F3+, +%284%2Api%29%2F3+, +pi%2F4+, +%285%2Api%29%2F4+
If you can, get another opinion on this also