Question 1117291
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<pre>
Use the formula

sin(A-B) = sin(A)*cos(B) - cos(A)*sin(B).     (*)


For it, in addition to the given values  sin(A) = {{{4/5}}} and cos(B) = {{{-5/13}}}  you need to find  cos(A)  and  sin(B).


1.  Since sin(A) = {{{4/5}}},  cos(A) = {{{sqrt(1 - sin^2(A))}}} = {{{sqrt(1-(4/5)^2)}}} = {{{sqrt(1-16/25)}}} = {{{sqrt(25-16)/25)}}} = {{{sqrt(9/25)}}} = {{{3/5)}}}.

    Notice that cos(A) is positive in QI.



2.  Since cos(B) = {{{-5/13}}},  sin(B) = {{{sqrt(1 - cos^2(B))}}} = {{{sqrt(1-(-5/13)^2)}}} = {{{sqrt(1-25/169)}}} = {{{sqrt(169-25)/169)}}} = {{{sqrt(144/169)}}} = {{{12/13)}}}.

    Notice that sin(B) is positive in QII.



3.  Now you have everything to use the formula (*). Substitute all given and found values into (*). You will get  

    sin(A-B) = sin(A)*cos(B) - cos(A)*sin(B) = {{{(4/5)*(-5/13) - (3/5)*(12/13)}}} = {{{-20/65 - 36/65}}} = {{{-56/65}}}.


<U>Answer</U>.  &nbsp;sin(A-B) = {{{-56/65}}}.
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To see many other similar solved problems, look into the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Calculating-trigonometric-functions-of-angles.lesson>Calculating trigonometric functions of angles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Selected-problems-from-the-archive-on-calculating-trig-functions-of-angles.lesson>Advanced problems on calculating trigonometric functions of angles</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Evaluating-trigonometric-expressions.lesson>Evaluating trigonometric expressions</A> 

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

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