since the cosine is the adjacent over the hypotenuse,
and we are given that the cosine of x is 8 over 17, we
will draw a right triangle with an adjacent side of 8
and a hypotenuse of 17. Then the angle in that
triangle that has 8 for its adjacent side will be
angle x.
Since the tangent is the opposite over the adjacent,
and we are given that the tangent of y is 5 over 12,
we will draw a right triangle with an opposite side
of 5 and an adjacent side of 12. Then the angle in
that triangle that has 5 for its opposite side and
12 for its adjacent side will be angle y.
Now we will find the missing side in each of those
triangles using the Pythagorean theorem:
c² = a² + b² c² = a² + b²
17² = 8² + b² c² = 12² + 5²
289 = 64 + b² c² = 144 + 25
225 = b² c² = 169
15 = b c = 13
Now we use the identity
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
and use the sides of those triangles to substitute
the trig ratios:
sin(x+y) = = =
Edwin
