Expert: Steve Holleran - 1/4/2008

Question
I'm sorry to bother you again. But can you help me with these too. I need to finish my trig course asap and I don't understand the convoluted language of the lesson.

9. Arccot (-0.555)

12. 2cos(x) - 2 = -1
Just checking: I got this: 120° + 360°n, 240° + 360°n

13. sec(x) + 4 = 2

14. sin(x)cos(x)=0, where 0 < x <2pi

15. sin2(x) - 5 = cos2(x) - 1

16. sin2(x) + sin(x) = 2, where 0 < x <2pi
I got this: 2pi/3, + 2pin

17. cos(1/2x) = 1/2

18. 4cos2(x) = 3

19. sin(x) = 2 cos2(x) - 1

20. sin2(2x) = 1

21. 4 sin(x)cos(x) - 2sin(x) - 2cos(x) = -1, where 0 < x <2pi

Thank you in advance for your help.

Answer
Hi Sana,

Whew! This is a lot to ask all at once. Can you try to send them in smaller groups in the future?

Anyway, let's see:

9.  If Arccot(-0.555) = x, then cot x = -.555 = 1/ tan x

so tan x = 1/-.555 and x = Arctan(1/-.555) = -1.06 radians.

12.  2 cos x - 2 = -1 -----> 2 cos x = 1

    so cos x = 1/2 ---> x = 60 + 360n ; 300 + 360n

13.  sec x + 4 = 2  ----> sec x = -2 --> cos x = -1/2

    so x = 120 + 360n ; 240 + 360n

14.  if sin x * cos x = 0 then either

    sin x = 0 -----> x = 0, pi

    cos x = 0 ---->  x = pi/2, 3pi/2

15.  Here, if you rearrange, you can get:

    sin 2x - cos 2x = 4

    Since sin 2x at most can = 1, and cos 2x at least can
    = -1, the largest difference can be 2.  So here there
   is no solution.

16.  sin 2x + sin x = 2

    Again, since sine values have a max of 1, this will
    only happen if sin 2x = 1 AND sin x = 1.  But if sin x = 1, then x = pi/2, which makes 2x = pi, and the sin pi = 0,
so here there is no solution.

17.  cos (1/2 x) = 1/2--> 1/2 x = 60  or 1/2 x = 300

which makes x = 120  +/- 360n  or x = 600 +/- 360n

18.  4 cos 2x = 3 ---> cos 2x = 3/4  --> 2x = 0.7227 rad

  so x = .3614 radians

19.  sin x = 2 cos 2x - 1

    sin x + 1 = 2 cos 2x --> sin x + 1 = 2(1 - 2 sin^2 x)

    sin x + 1 = 2 - 4 sin^2 x

    4 sin^2 x + sin x - 1 = 0

    Now use the quadratic formula to get

    sin x = .3903 ----> x = Arcsin(.3903) = .4009 rad

    sin x = -.6403 -->  x = Arcsin(-.6403) = -.6949 rad

20.  sin 2(2x) = 1 ----> sin 4x = 1

    4x = pi/2 ; 5pi/2; 9pi/2 ; 13pi/2  etc

     x = pi/8 ; 5pi/8 ; ......

21.   4 sin x cos x - 2 sin x - 2 cos x = -1

     4 sin x cos x - 2 sin x - 2 cos x + 1 = 0

     now factor by grouping:

     2 sin x(2 cos x - 1) - 2 cos x + 1 = 0

     2 sin x(2 cos x - 1) -1(2 cos x - 1) = 0

     (2 cos x - 1)(2 sin x - 1) = 0

     so either 2 cos x - 1 = 0 --> cos x = 1/2

                                   x = pi/3, 5pi/3

    or 2 sin x - 1 = 0 --> sin x = 1/2

                           x = pi/6 , 5pi/6

Hope this helps you out, and that you finish your trig course satisfactorily.

Steve