Make sketches of two triangles with the given data, both must be
in quadrant II
sin a = 2/3 = opposite/hypotenuse
you triangle is right-angled with y = 2 and r = 3
x^2 + y^2 = r^2
x^2 + 4 = 9
x = ±√5, but from the given domain, we know x = -√5
so cos a = -√5/3
for the 2nd triangle, cos b = -1/5, so x, the adjacent = -1, hypotenuse = 5
(-1)^2 + y^2 = 25
y = √24 and sin b = √24/5 or 2√6/5
then sin(a+b) = (sina)(cosb) + cosa(sinb)
=(2/3)(-1/5) + (-√5/3)(2√6/5)
= -2/15 - 2√30/15
= -(2 + 2√30)/15
from you expansion of cos(a - b)
follow the same steps, you already know the sines and cosines of a and b
Given that sin(a) = 2/3 and cos(b) = −1/5, with a and b both in the interval [𝜋/2, 𝜋), find the exact values of sin(a + b) and cos(a − b).
1 answer