SOLUTION: 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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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
Answer by ikleyn(52797)   (Show Source): You can put this solution on YOUR website!
.
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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

To evaluate    you will use the formula 

 = .     (1)


To use this formula, you need to know  ,  ,    and  .

Is it given to you? - Only in part: you are given  " =   and   is in Q4".

Regarding  , you are given that  " =   and   is in Q3".


So, your first task is, based on the given data, find  ,    and  .

You have   =  =  =  =  = .

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


Having  = , you can easily calculate 

 =  =  =  =  =  = .

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


Next,   =  =  =  =  = .

   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:   = ,    = ,   =   and   = .

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

 =  =  =  = . 

Answer.   = .

For many other similar problems see the lessons
    - Calculating trigonometric functions of angles
    - Advanced problems on calculating trigonometric functions of angles
in this site.


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