Question 38936
<pre><font size = 4><b>Let <font face = "symbol">a</font> and <font face = "symbol">b</font> be acute angles with sin<font face = "symbol">a</font> = 4/5 and sin<font face = "symbol">b</font> = 5/13.
Find cos(<font face = "symbol">a</font> - <font face = "symbol">b</font>)

cos(<font face = "symbol">a</font> - <font face = "symbol">b</font>) = cos<font face = "symbol">a</font>·cos<font face = "symbol">b</font> + sin<font face = "symbol">a</font>·sin<font face = "symbol">b</font>

We need the cosines of <font face = "symbol">a</font> and <font face = "symbol">b</font>, so 
we draw right triangles:

Since sin<font face = "symbol">A</font> = 4/5, and since sine = opposite/hypotenuse, we
place the numerator 4 on the opposite side to angle <font face = "symbol">a</font> and
the denominator 5 on the hypotenuse

    /|
 5 / |4
  /  |
 /<u><font face = "symbol">a</font>  |</u>
 
Now we calculate the adjacent side using the Pythagorean
theorem. 

a² + b² = c²
a² + 4² = 5²
a² + 16 = 25
     a² = 9
      a = 3
     
    /|
 5 / |4
  /  |
 /<u><font face = "symbol">a</font>  |</u>
   3

So cos<font face = "symbol">a</font> = 3/5

Since sin<font face = "symbol"></font> = 5/13, and since sine = opposite/hypotenuse, we
place the numerator 5 on the opposite side to angle <font face = "symbol">a</font> and
the denominator 13 on the hypotenuse

    /|
13 / |5
  /  |
 /<u><font face = "symbol">b</font>  |</u>
 
Now we calculate the adjacent side using the Pythagorean
theorem. 

a² + b² = c²
a² + 5² = 13²
a² + 25 = 169
     a² = 144
      a = 12
     
    /|
13 / |5
  /  |
 /<u><font face = "symbol">b</font>  |</u>
  12
  
So cos<font face = "symbol">b</font> = 12/13

Now we have

sin<font face = "symbol">a</font> = 4/5, sin<font face = "symbol">b</font> = 5/13, cos<font face = "symbol">a</font> = 3/5, cos<font face = "symbol">b</font> = 12/13

Substitute into

cos(<font face = "symbol">a</font> - <font face = "symbol">b</font>) = cos<font face = "symbol">a</font>·cos<font face = "symbol">b</font> + sin<font face = "symbol">a</font>·sin<font face = "symbol">b</font>

cos(<font face = "symbol">a</font> - <font face = "symbol">b</font>) = (3/5)(12/13) + (4/5)(5/13)

cos(<font face = "symbol">a</font> - <font face = "symbol">b</font>) = 36/65 + 20/65

cos(<font face = "symbol">a</font> - <font face = "symbol">b</font>) = 56/65         

Edwin
AnlytcPhil@aol.com</pre>