i need trigonometry equation help pronto!!!?

Question

how do you solve these:

1. 8cotx=-7 (between 0 and 360 degrees)
2. cscx=-12 (between 0 and 360 degrees)
3. sin(3x)=.8 (all real answers)
4. csc(x/4) = -square root 2 (between 0 and 2pi)

Answer 1

The usual way, doing your math right, knowing where the functions are positive and negative, knowing how they behave, and knowing how to handle arc-functions.

In the first, we isolate the trig function so we have
cot(x)= -7/8, taking "inverses" cot(-7/8)=x This is read as "x is the angle whose cotangent is -7/8."
So where is the cotangent negative? Since it is reciprocal tangent, it is negative where ever the tangent is negative, which is in the 2 and 4 quadrants. We can compute the cotangent of 7/8 in our friendly calculator or with a trig table; it is about 48 degrees. By the behavior of the cotangent, the 48 degrees is reckoned from the end of the quadrants MINUS the 48 degrees. So the answers would be 180-48 and 360-48 or
132 and 312 degrees appx.

Prob 3 is interesting, so lets try it. We don't have to do any extra math now, so we can take inverses and get arcsin(.8)=3x. This means, 3x is the angle whose sine is 0.8. Again, to the calculator or table, and this is about 54 degrees. So x is 18 degrees. Are we thru? No, it asks for all REAL answers. Since we have a periodic function, we have to see what happens in the first full cycle and generalize it to subsequent cycles. The sine is positive in the second quadrant, and from the behavior of the sine, the sine in the second quadrant is the mirror image of the sine in the first quadrant, so the angle 3x is 180 MINUS 54 degrees or 126 degrees, so x is 42 degrees.
To generalize, we write the answer as
x = 360(n-1)+18 and 360(n-1)+42, where n can be any positive integers.

Answer 2

1. 8cotx=-7
cot x = -7/8............Isolate cot x first
tan x = -8/7

First of all, we must determine where the angle is. Since it is negative, where is tan negative? Tan is negative in the 2nd and 4th quadrants. So, what I would do, is on a piece of paper draw a diagonal line on the 2nd and 4th quadrant, indicating the angle.

Now, whenever we have a variable for the angle, we must always find the reference angle before we do anything else.

Reference angle x = inverse tan(8/7)
We do not put the negative in the (8/7) for the reference angle.

Once you work that out, your

Reference Angle = 48.8 deg.

Alright, now on your piece of paper where you drew your two diagonal lines on the 2nd and 4th quadrant, lable the reference angles. Remember, you reference angles are found in between the x-axis and the diagonal line.

You can now find what angle x is

x1 = 180 - 48.8
= 131. 2 deg

x2 = 360 - 48.8
= 311.2 deg

Answer 3

remember that an equation can be modified

such that Ex: x= arcsin use that as help