math prob(sine cosine tangent)

1)
Any angles U, V such that U ≠ nπ (n is an integer) and V = - U. Proof: cos (U) = cos(-U) (Law of Cosine)= cos (V) (since V= -U)Therefore, cos(U)= cos(V)sin(V)= sin(-U) (since V-U) = -sin (U)Since sine is an odd function, so sin(U) ≠ sin (-U) for any angle U≠ nπ. Therefore, sin(U)≠sin(V)

2)
e^sinx = ¼ ln e^sinx = ln(¼)sinx = ln1-ln4sinx = 0 – ln4sinx = -ln4Since |sinx|≤1 for all real number x and –ln4<1, therefore there doesn’t exist a real number x that will satisfy the equation.
4)
Suppose x is any real number such that cos x ≠0 Since |cos x|≤1, therefore 1 ≤ 1/|cos x||sin x| ≤ |sin x|/|cos x||sin x|≤ |sin x / cos x| (Note: tan x = sin x / cos x)So |sin x|≤|tan x|
5) cos is an even function since cos (x) = cos (-x) for all real x numbers!
6)
cos 85º+cos95º = cos(90º-5º)+cos(90º+5º) =(cos90cos5+sin90sin5)+(cos90cos5-sin90sin5) (law of cosine)= (0+sin5)+(0-sin5) (cos 90 = 0)= 0Therefore, cos 85º+cos95º = 0

I hope this will help. I have to think about number 3. The other answers should be easy to understand!

2011-03-29 09:41:45 補充：
2) typo, it should be
Since |sinx|≤1 for all real number x and –ln4< -1, therefore there doesn’t exist a real number x that will satisfy the equation.

2011-03-30 08:33:52 補充：
3) Since |cos x|<=1, therefore (cos x)^99 <=1 and 4cos x 4 <= 4.
So (cos x)^99 + 4cos x<= 5 which implies (cos x)^99 + 4cos x - 6 <= -1. As a result,
there is no solution for (cos x)^99 + 4cos x - 6 = 0 for all x.

1) cos 30 = cos (-30), but sin 30 is not equal to sin (-30).
2) Since -1 < sin x < 1, min. value of e^(sin x) = e^(-1) = 1/e = 0.36 which is greater than 0.25, so no real value of x for e^(sin x) = 0.25.
3) Max. value of (cos x)^99 = 1, max. value of 4 cos x = 4, max. value of (cos x)^99 + 4 cos x = 1 + 4 = 5, so cannot be equal to 6. So no solution.
4) sin x = opposite side/hypotenuse, tan x = opposite side/adjacent side, since hypotenuse > adjacent side for all angles except 0, so sin x < tan x for all angles except 0.
5) cos x = cos (-x), so it is an even function.
6) cos 85 = cos (90 - 5) = sin 5.
cos 95 = cos (90 + 5) = - sin 5, so their sum = 0.