SOLUTION: Solve the equation on the interval [o,2pi) 1.) cos(2theta - pi/2)= -1 I started off by adding pi/2 to -1 so my new equation is cos2theta=-1/2 Not sure if this is correct but

Algebra ->  Trigonometry-basics -> SOLUTION: Solve the equation on the interval [o,2pi) 1.) cos(2theta - pi/2)= -1 I started off by adding pi/2 to -1 so my new equation is cos2theta=-1/2 Not sure if this is correct but      Log On


   

Question 794776: Solve the equation on the interval [o,2pi)
1.) cos(2theta - pi/2)= -1
I started off by adding pi/2 to -1 so my new equation is cos2theta=-1/2
Not sure if this is correct but I'm stuck there.

2.) tan^2theta=3/2sec theta
I'm completely lost on this probem

Thank you in advance!
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
1.) +cos%282theta+-+pi%2F2%29=+-1+
There is only one angle in the %22%5B0%2C%222pi%22%29%22 interval that has
cox%28x%29=-1:
x=pi cos%28pi%29=-1
So 2theta-pi%2F2=pi-->2theta=pi%2Bpi%2F2-->2theta=3pi%2F2-->highlight%28theta=3pi%2F4%29

2.) tan%5E2%28theta%29=%283%2F2%29sec%28theta%29
tan%28theta%29=sin%28theta%29%2Fcos%28theta%29 --> tan%5E2%28theta%29=sin%5E2%28theta%29%2Fcos%5E2%28theta%29
Also sec%28theta%29=1%2Fcos%28theta%29
So we can re-write the equation as
+sin%5E2%28theta%29%2Fcos%5E2%28theta%29+=%283%2F2%29%281%2Fcos%28theta%29%29
Multiplying both sides times cos%5E2%28theta%29 we get
+sin%5E2%28theta%29+=%283%2F2%29cos%28theta%29
Multiplying times 2, we get
2+sin%5E2%28theta%29+=3cos%28theta%29
Since sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1-->sin%5E2%28theta%29=1-cos%5E2%28theta%29
so we can re-write the equation as
2%281-cos%5E2%28theta%29%29=3cos%28theta%29-->2-2cos%5E2%28theta%29=3cos%28theta%29 -->2cos%5E2%28theta%29%2B3cos%28theta%29-2=0
Since I'm tired of writing cos%28theta%29 so many times, I will temporarily rename it
+x+=+cos%28theta%29
Now my equation looks like 2x%5E2%2B3x-2=0
The solutions to that equation are x=-2 and x=1%2F2
We can only use x=1%2F2-->cos%28theta%29=1%2F2 because -1%3C=cos%28theta%29%3C=1.
If cos%28theta%29=1%2F2 and 0%3C=theta%3C2pi
then highlight%28theta=pi%2F3%29 or highlight%28theta=5pi%2F3%29