Question 1179362

Find the exact value of the trigonometric function given that sin u = -12/13 and cos v = 24/25. (Both u and v are in Quadrant IV) a)sin (u+v) b)cos(v-u) c)tan(u-v)
<pre>{{{matrix(1,7, sin (u), "=", (- 12)/13, "=", O/H, "=", y/r)}}}. With y = - 12, and r = 13, x = 5, since this represents a 5-12-13 PYTHAG. Triple. We also get: {{{matrix(2,7, cos (u), "=", A/H, "=", x/r, "=", 5/13, tan (u), "=", O/A, "=", y/x, "=", (- 12)/5)}}}
{{{matrix(1,7, cos (v), "=", 24/25, "=", A/H, "=", x/r)}}}. With x = 24, and r = 25, y = - 7, since this represents a 7-24-25 PYTHAG. triple. We also get: {{{matrix(2,7, sin (v), "=", O/H, "=", y/r, "=", (- 7)/25, tan (v), "=", O/A, "=", y/x, "=", (- 7)/24)}}}
<b>a)</b> {{{highlight_green(matrix(5,3, sin (u + v), "=", sin (u) cos (v) + cos (u) sin (v), sin (u + v), "=", (- 12/13)(24/25) + (5/13)(- 7/25), sin (u + v), "=", (- 12)(24)/13(25) + (5)(- 7)/13(25), sin (u + v), "=", (- 288)/325 + (- 35)/325, sin (u + v), "=", (- 323)/325))}}}     <b>b)</b> {{{highlight_green(matrix(5,3, cos (v - u), "=", cos (v) cos (u) + sin (v) sin (u), cos (u - v), "=", (24/25)(5/13) + ((- 7)/25)(- 12/13), cos (u - v), "=", 24(5)/25(13) + (- 7)(- 12)/25(13), cos (u - v), "=", 120/325 + 84/325, cos (u - v), "=", 204/325))}}} 
                                               <b>c)</b> {{{highlight_green(matrix(5,3, tan (u - v), "=", (tan (u) - tan (v))/(1 + tan (u) tan (v)), tan (u - v), "=", ((- 12)/5 - (- 7/24))/(1 + (- 12/5)(- 7/24)), tan (u - v), "=", ((- 12/5) + (7/24))/(1 + (- 12)(- 7)/5(24)),
tan (u - v), "=", (((- 12)(24) + (7)(5))/5(24))/(5(24)/5(24) + (- 12)(- 7)/5(24)), tan (u - v), "=", ((- 288 + 35)/5(24))/((120 + 84)/(5(24))))))}}}
                                                   {{{matrix(1,3, tan (u - v), "=", (- 288 + 35)/5(24))}}} ÷ {{{(120 + 84)/5(24)}}}
                                                   {{{matrix(2,5, tan (u - v), "=", (- 253)/5(24), "*", 5(24)/204, tan (u - v), "=", (- 253)/cross(5(24)), "*", cross(5(24))/204)}}}
                                                    {{{highlight_green(matrix(1,3, tan (u - v), "=", (- 253)/204))}}}