Question 1163852

Find the exact value of the trigonometric function below given that sin u = -7/25 and cos v = -4/5. (Both u and v are in Quadrant III).

cos (u-v)
<pre>Difference of 2 angles formula: cos (A  -  B) = cos A cos B + sin A sin B
This gives us: cos (u  -  v) = cos u cos v + sin u sin v
{{{matrix(1,7, sin (u), "=", O/H, "=", (- 7)/25, "=", y/r)}}}
The above represents a "7-24-25" PYTHAG TRIPLE, and so, x = 24.
However, because the opposite side (y), and the adjacent side (x), are in the 3rd quadrant, then y = O = - 7, and x = A = - 24.
Therefore, {{{matrix(1,7, cos (u), "=", A/H, "=", x/r, "=", (- 24)/25)}}}

{{{matrix(1,7, cos (v), "=", A/H, "=", (- 4)/5, "=", y/r)}}}
The above represents a "3-4-5" PYTHAG TRIPLE, and so, x = 3.
However, because the opposite side (y), and the adjacent side (x), are in the 3rd quadrant, then y = O = - 4, and x = A = - 3.
Therefore, {{{matrix(1,7, sin (v), "=", O/H, "=", x/r, "=", (- 3)/5)}}}

We now get: {{{matrix(4,3, cos (u - v), "=", cos (u) cos (v) + sin (u) sin (v), 
cos (u - v), "=", (- 24/25) * ((- 4)/5) + (- 7/25) * ((- 3)/5),
cos (u - v), "=", (- 24)(- 4)/(25 * 5) + (- 7)(- 3)/(25 * 5),
cos (u - v), "=", 96/125 + 21/125)}}}
            {{{highlight_green(matrix(1,3, cos (u - v), "=", highlight(117/125)))}}}</pre>