SOLUTION: find the exact value of sin(x-y) if sin x = 4/9 and sin y = 1/4

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Question 1154735: find the exact value of sin(x-y) if sin x = 4/9 and sin y = 1/4

Found 3 solutions by greenestamps, MathLover1, ikleyn:
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


There is not enough information to find the exact value.

sin%28x-y%29+=+%28sinx%29%28cosy%29-%28cosx%29%28siny%29

Given sinx = 4/9, we know cosx is EITHER sqrt(65)/9 OR -*sqrt(65)/9;
given siny = 1/4, we know cosx is EITHER sqrt(15)/4 OR -sqrt(15)/4.

Since there is no information allowing us to determine which values to use for cosx and cosy, we can't find the value of sin(x-y).

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

the exact value of
sin%28x-y%29 if sin%28x%29+=+4%2F9 and sin%28y%29+=+1%2F4
use identities:
sin%28x-y%29=sin%28x%29+cos%28y%29+-+cos%28x%29+sin%28y%29
and
cos%5E2%28x%29=1-sin%5E2%28x%29

if sin%28x%29+=+4%2F9, then cos%5E2%28x%29=1-%284%2F9%29%5E2=1-16%2F81=65%2F81
=> cos%28x%29=sqrt%2865%2F81%29=sqrt%2865%29%2F9
sin%28y%29+=+1%2F4
cos%5E2%28y%29=1-%281%2F4%29%5E2
cos%5E2%28y%29=1-1%2F16
cos%5E2%28y%29=15%2F16
cos%28y%29=sqrt%2815%2F16%29
cos%28y%29=sqrt%2815%29%2F4
Therefore,
sin%28x-y%29=%284%2F9%29+%28sqrt%2815%29%2F4%29+-+%281%2F4%29%28sqrt%2865%29%2F9%29+
sin%28x-y%29=+sqrt%2815%29%2F9+-+sqrt%2865%29%2F36+
sin%28x-y%29=+4sqrt%2815%29%2F36+-+sqrt%2865%29%2F36+
sin%28x-y%29=+%284sqrt%2815%29+-+sqrt%2865%29%29%2F36+-> the exact value

Answer by ikleyn(52814) About Me  (Show Source):
You can put this solution on YOUR website!
.

In order for this problem (and many other similar problems) be correctly posed, it MUST say to which quarter
every of the angles x and y do belong.