Expert: Richard J. Raridon - 1/23/2005

Question
sorry, i forgot to include this with the problem

0 < x < (pi/2)

does that mean "x" has to be in radian form, or can i just leave it in degree form.

also there was another problem i had csc(x) = 5, (pi/2) < x < (3/2)pi
when i solve for "x", if pi/2 is suppose to be 90, then "x" is lower than that, but if "pi" is suppose to mean the (22/7) value / 2, which becomes (11/14), then its still lower than (pi/2), does this mean that there is way to solve.

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Followup To
Question -
If tan(x) = -5/12
and
tan(x) also equals (sin(x))/(cos(x)), then how come sin(x) does't equal ±5 and cos(x) doesn't equal ±12. I know if you try to inverse cos or sin either one and you gives you an error.

I understand the rest you told me, if you can only explain that.

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Followup To
Question -
I need to check something. When i click cos^-1 on my computer is that giving me the value for Sec, same for the others. In otherwords, does

cos^-1 x = Sec x
sin^-1 x = Csc x
tan^-1 x = Cot

also if i have something like this

cot(x) = -12/5
do i just calculate -12/5, then click tan, since cot(x) and tan(x) are the opposite of each other.

is this correct to say

cot(x) = -12/5
tan(x) = -5/12

then what would i do to get sin and cos

The reason i am asking is because my nephew has asked me this question before, but because i haven't had a trig problem like this in like 5 or 6 years, i am having trouble remembering how to do this.

He gave me a few problems, and has asked me to find the other 5 functions.

Here is just one.

cot(x) = -12/5
Now i have to find the values of the other 5 functions.
Answer -
cos^-1(x) does not equal sec(x)
that's [cos(x)]^-1
when you put in a number and hit cos^-1(x)(called arc cosine), it gives you the angle for that value of cosine.  For example, if you put in 0.5 and hit cos^-1, you will get 60 degrees.
if you have cot(x) = -12/5, then tan(x) = -5/12 or -0.4167.
If you put in -0.4167 and hit tan^-1, you'll get -22.6 degrees, or 337.4 degrees.  
cos(-22.6) = 0.923, sin(-22.6) = -0.384
sec(-22.6) = 1.083, csc(-22.6) = -2.602
Hope that helps
Answer -
Draw a right triangle with sides x and y and hypotenuse r.
Consider the angle that is opposite side y, call it A.  
The sine of A is y/r and the cosine of A is x/r.  You should be able to see that neither sine nor cosine can exceed the value 1.0.  

Answer
If you have 0 < x < pi/2, that generally means to give the answer in radians.  
So for csc(x) = 5, pi/2 < x < 3pi/2
that means x has to be greater than pi/2 and less than 3pi/2
If csc(x) = 5, then sin(x) = 1/5 and x = 11.54 degrees.  
since that's less than pi/2, you have to find another angle whose sine = 0.2 and that's 180-11.54 or 168.46 degrees.  
If you want to change that to radians, divide by 57.3 to
get 2.94 radians