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(52794) About Me  (Show Source):
You can put this solution on YOUR website!
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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  sin%28theta-phi%29  you will use the formula 

sin%28theta-phi%29 = sin%28theta%29%2Acos%28phi%29+-+cos%28theta%29%2Asin%28phi%29.     (1)


To use this formula, you need to know  sin%28theta%29,  cos%28phi%29,  cos%28theta%29  and  sin%28phi%29.

Is it given to you? - Only in part: you are given  "sin%28phi%29 = -sqrt%2810%29%2F10  and  phi is in Q4".

Regarding  theta, you are given that  "tan%28theta%29 = 12%2F5  and  theta is in Q3".


So, your first task is, based on the given data, find  sin%28theta%29,  cos%28theta%29  and  cos%28phi%29.

You have  sin%28theta%29 = -sqrt%281%2F%281+%2B+1%2Ftan%5E2%28theta%29%29%29 = -sqrt%28+1%2F%281+%2B+%285%2F12%29%5E2%29%29 = -sqrt%281%2F%28%281%2B+25%2F144%29%29%29 = -sqrt%28144%2F169%29 = -12%2F13.

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


Having sin%28theta%29 = -12%2F13, you can easily calculate 

cos%28theta%29 = -sqrt%281+-+sin%5E2%28theta%29%29 = -sqrt%281+-+%28-12%2F13%29%5E2%29 = -sqrt%281+-+144%2F169%29 = -sqrt%28%28169-144%29%2F169%29 = -sqrt%2825%2F169%29 = -5%2F13.

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


Next,  cos%28phi%29 = sqrt%281+-+sin%5E2%28phi%29%29 = sqrt%281+-+%28-sqrt%2810%29%2F10%29%5E2%29 = sqrt%281+-+%2810%2F100%29%29 = sqrt%2890%2F100%29 = %283%2Asqrt%2810%29%29%2F10.

   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%28theta%29 = -12%2F13,   cos%28phi%29 = %283%2Asqrt%2810%29%29%2F10,  cos%28theta%29 = -5%2F13  and   sin%28phi%29= -sqrt%2810%29%2F10.

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

sin%28theta-phi%29 =  = -%2812%2A3%2Asqrt%2810%29%29%2F130+-+%285%2Asqrt%2810%29%29%2F130 = -%28%2836%2B5%29%2Asqrt%2810%29%29%2F130 = -%2841%2Asqrt%2810%29%29%2F130. 

Answer.  sin%28theta-phi%29 = -%2841%2Asqrt%2810%29%29%2F130.

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