Expert: Paul Klarreich - 10/12/2007

Question
*Find cscx if cosx=2sqrt6/7 and x lies in quadrant I
specific number answer
*Find cos(2x) if sinx=7/5 and x lies in quadrant I
specific number answer
*cscx(sinx + cosx)
identity answer

Answer
Questioner:   Danixa
Category:  Calculus
Private:  No
 
Subject:  Pre-Calculus
Question:  *Find cscx if cosx=2sqrt6/7 and x lies in quadrant I
specific number answer
*Find cos(2x) if sinx=7/5 and x lies in quadrant I
specific number answer
*cscx(sinx + cosx)
identity answer
-----------------------------------------
Hi, Danixa,

Here is the general way to handle things like this :  you have the value of one of the trigonometric functions and you want one or more of the others:

You wrote:  cosx=2sqrt6/7

First, don't use 'x' as the name of the angle, because we use it for something else.  We will use 't' which stands for 'theta'.

        2 sqrt(6)    x
cos t =  --------- = ---
            7        r

Remember,  x/r is the definition of  cos t.

Now just set  x = 2 sqrt(6)  and  set r = 7.  Then use the fact that

x^2 + y^2 = r^2

to find the third number, y.

(2 sqrt(6))^2 + y^2 = (7)^2

4(6) + y^2 = 49

24 + y^2 = 49

y^2 = 25,  y = 5.

Actually, we should be writing  y = +- 5.  In this case we are told that theta lies in quadrant 1, so we choose  y = 5.

Now csc t = r/y,
         7
csc t = -----
         5
.................................
*Find cos(2x) if sinx=7/5 and x lies in quadrant I

Since the conditions here are impossible, I'll leave this one to you.  Suggestion:

(1) fix the error.  Maybe you meant  sin x = 5/7.
(2) Use  cos(2x) = 1 - 2 sin^2(x)

If that is the case, you would have:

cos(2x) = 1 - 2(25/49) = 1 - 50/49 = -1/49

.........................

*cscx(sinx + cosx)
identity answer  << what does this mean?