Please can you tell me how to solve the following equations for values of x between (0.2 π) {π is pie}
a) tan (4x) = 1
b) sin(x-2π) + sin (x - 2π) = ½
c) 2 cos (x + ½) = -1
d) sin x + cos x = 3/2
Well when you want to isolate a variable that's inside a trig function, you just take the inverse trig function.
For instance, with the first problem:
.
We take of both sides (sometimes, is written and pronounced ). Get:
, so resolving the left side gives:
.
We now have to figure out what is. Well it's like asking "what angle in a right triangle gives a ratio of opposite side over adjacent side equal to 1?" It should be obvious that a 45-45-90 triangle passes this test, so resolves to , or . However, there may be more answers. Notice that works as well. In fact, for any integer n, , so could be any of these.
We take this info and go back to our equation:
. So:
. Then:
.
All right, but you were given a restriction of possible values, .
So we want to figure out the only values of n that make our solution valid. ,
,
,
.
So n could only be 0,1,2,3,4,5,6, or 7.