How do you find the exact value $cos(x+y)$ if $tanx=5/3,siny=1/3$ ?

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Answer 1 · 2017-01-17T19:49:11

$cos(x+y)=(6sqrt2-5)/(3sqrt34)$

Use the cosine angle addition formula:

$cos(x+y)=cosxcosy-sinxsiny$

We need to determine $sinx$ , $cosx$ , and $cosy$ from whatever we have.

Determining $mathbf(sinx,cosx: )$

$tan^2x+1=sec^2x" "" "" "$ use $tanx=5/3$

$25/9+1=sec^2x$

$secx=sqrt(34/9)=sqrt34/3$

Then:

$color(blue)(cosx=1/secx=3/sqrt34$

Now using:

$sin^2x+cos^2x=1" "" "" "$ where $cosx=3/sqrt34$

$sin^2x+9/34=1$

$color(blue)(sinx=sqrt(25/34)=5/sqrt34$

Determining $mathbf(cosy: )$

$sin^2y+cos^2y=1" "" "" "$ use $siny=1/3$

$1/9+cos^2y=1$

$color(blue)(cosy=sqrt(8/9)=(2sqrt2)/3$

Returning to $mathbf(cos(x+y): )$

$cos(x+y)=cosxcosy-sinxsiny$

$color(white)(cos(x+y))=3/sqrt34((2sqrt2)/3)-5/sqrt34(1/3)$

$color(white)(cos(x+y))=(6sqrt2-5)/(3sqrt34)$

Note these are all assuming that $x$ and $y$ are in the first quadrant (everything is positive).

How do you find the exact value cos(x+y)cos(x+y) if tanx=5/3,siny=1/3tanx=5/3,siny=1/3?