How do you use double-angle identities to find the exact value of sin 2x & cos 2x when csc(x) = -25/7 and cos x>0?

1 Answer
Apr 23, 2018

Find #sinx# first and then find #sin2x# and #cos2x#.

Explanation:

As #cos x# > 0, the basic angle must be in quadrant II or III.
As #csc x# = #-25/7#, the basic angle must be in quadrant III as #csc x# = #1/sin x#.

By algebra, #sin x# = #-7/25#. From there, you can draw the triangle and find out what the basic angle is.
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#sin 2x# = #2sin x cos x#
#cos 2x# = #1- 2sin^2x# (there are other variations but this works better for this question)

From the triangle, we can find that #cos x = -24/25#
The negative sign is because the basic angle is in the third quadrant, where only tangent is positive,

Evaluating #sin 2x# and #cos 2x#, #sin2x# = #-336/625# and #cos2x# = #-527/625#.

The negative signs are because the angles fall in the third quadrant. Not so sure about that though.