SOLUTION: If sin α = 4/5 and cos β = -5/13 for α in Quadrant I and β in Quadrant II, find cos(α - β).

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Question 1138000: If sin α = 4/5 and cos β = -5/13 for α in Quadrant I and β in Quadrant II, find cos(α - β).
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.
If sin α = 4/5 and cos β = -5/13 for α in Quadrant I and β in Quadrant II, find cos(α - β).
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Use the formula 

    cos(a-b) = cos(a)*cos(b) + sin(a)*sin(b)    (1)


Regarding this formula, see the lesson Addition and subtraction formulas in this site.


In addition to the given  sin(a) = 4%2F5  and  cos(b) = -5%2F13, you need to know  cos(a) and sin(b).



    1.  cos(a) = sqrt%281-sin%5E2%28a%29%29 = sqrt%281+-+%284%2F5%29%5E2%29 = sqrt%281+-+16%2F25%29 = sqrt%28%2825-16%29%2F25%29 = sqrt%289%2F25%29 = 3%2F5.

        The sign "+" was chosen at the square root because the angle "a" is in QI.


    2.  sin(b) = sqrt%281-cos%5E2%28b%29%29 = sqrt%281+-+%28-5%2F13%29%5E2%29 = sqrt%281+-+25%2F169%29 = sqrt%28%28169-25%29%2F169%29 = sqrt%28144%2F169%29 = 12%2F13.

        The sign "+" was chosen at the square root because sin(b) is positive when the angle "b" is in QII.



Now all you need to do is to substitute everything into the formula (1) and make the calculations.


cos(a-b) = %283%2F5%29%2A%28-5%2F13%29+%2B+%284%2F5%29%2A%2812%2F13%29 = -15%2F65+%2B+48%2F65 = %28-15%2B48%29%2F65 = 33%2F65.     ANSWER

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To see many other similar solved problems on calculating/evaluating trig functions,  look into the lessons
    - Calculating trigonometric functions of angles
    - Advanced problems on calculating trigonometric functions of angles
    - Evaluating trigonometric expressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Trigonometry: Solved problems".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.


Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
The formula is .

Draw angle α in Quadrant I:

Since sine = y%2Fr, we make y=4 and r=5, so that the sin(α)
will be y%2Fr%22%22=%22%224%2F5.



For the formula, we need sine and cosine, and the cosine is x%2Fr

So we find x by the Pythagorean relation: 

x%5E2%2By%5E2=r%5E2
x%5E2%2B%284%29%5E2=%285%29%5E2
x%5E2%2B16=25
x%5E2=9
x+=+%22%22+%2B-+3

Since x goes to the right, we know to take the positive
value x+=+%22%22%2B3.  So now we know that cos%28alpha%29=x%2Fr=3%2F5

Next we draw angle β in Quadrant II:

Since cosine = x%2Fr, we make x=-5 and r=13, so that the cos(β)
will be x%2Fr%22%22=%22%22%28-5%29%2F13.



For the formula, we need sine and cosine, and the sine is y%2Fr

So we find x by the Pythagorean relation: 

x%5E2%2By%5E2=r%5E2
%28-5%29%5E2%2By%5E2=%2813%29%5E2
25%2By%5E2=169
y%5E2=144
y+=+%22%22+%2B-+12

Since y goes up from the x-axis, we know to take the positive
value y+=+%22%22%2B12.  So now we know that sin%28beta%29=y%2Fr=12%2F13

Now we use the formula

.



matrix%281%2C3%2Ccos%28alpha-beta%29%2C%22%22=%22%22%2C%28-15%2F65%29%2B%2848%2F65%29%29

matrix%281%2C3%2Ccos%28alpha-beta%29%2C%22%22=%22%22%2C33%2F65%29

Edwin