Question 1036430
.
Evaluate the expression under the given conditions

sin(θ − ϕ); tan θ = 12/5, θ in Quadrant III, sin ϕ = -(sqrt 10)/10 , ϕ in Quadrant IV
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<pre>
To evaluate  {{{sin(theta-phi)}}}  you will use the formula 

{{{sin(theta-phi)}}} = {{{sin(theta)*cos(phi) - cos(theta)*sin(phi)}}}.     (1)


To use this formula, you need to know  {{{sin(theta)}}},  {{{cos(phi)}}},  {{{cos(theta)}}}  and  {{{sin(phi)}}}.

Is it given to you? - Only in part: you are given  "{{{sin(phi)}}} = {{{-sqrt(10)/10}}}  and  {{{phi}}} is in Q4".

Regarding  {{{theta}}}, you are given that  "{{{tan(theta)}}} = {{{12/5}}}  and  {{{theta}}} is in Q3".


So, your first task is, based on the given data, find  {{{sin(theta)}}},  {{{cos(theta)}}}  and  {{{cos(phi)}}}.

You have  {{{sin(theta)}}} = {{{-sqrt(1/(1 + 1/tan^2(theta)))}}} = {{{-sqrt( 1/(1 + (5/12)^2))}}} = {{{-sqrt(1/((1+ 25/144)))}}} = {{{-sqrt(144/169)}}} = {{{-12/13}}}.

   The sign "-" was chosen for the square root, because the sine  is negative for an angle in Q3.


Having {{{sin(theta)}}} = {{{-12/13}}}, you can easily calculate 

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

   The sign "-" was chosen for the square root, because the cosine  is negative for an angle in Q3.


Next,  {{{cos(phi)}}} = {{{sqrt(1 - sin^2(phi))}}} = {{{sqrt(1 - (-sqrt(10)/10)^2)}}} = {{{sqrt(1 - (10/100))}}} = {{{sqrt(90/100)}}} = {{{(3*sqrt(10))/10}}}.

   The sign "+" was chosen for the square root, because the cosine is positive for an angle in Q4.

Let us summarize what we have so far:  {{{sin(theta)}}} = {{{-12/13}}},   {{{cos(phi)}}} = {{{(3*sqrt(10))/10}}},  {{{cos(theta)}}} = {{{-5/13}}}  and   {{{sin(phi)}}}= {{{-sqrt(10)/10}}}.

Now you can plug-in this data into the formula (1) and get

{{{sin(theta-phi)}}} = {{{(-12/13)*((3*sqrt(10))/10) - (-5/13)*((-sqrt(10))/10)}}} = {{{-(12*3*sqrt(10))/130 - (5*sqrt(10))/130}}} = {{{-((36+5)*sqrt(10))/130}}} = {{{-(41*sqrt(10))/130}}}. 

<U>Answer</U>.  {{{sin(theta-phi)}}} = {{{-(41*sqrt(10))/130}}}.
</pre>

For many other similar problems see 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>

in this site.