suppose sinx=1/5, cosy=2/3, and x and y are in the first quadrant. determine sin (x+y)
To find sin(x + y), we need to use the sum of angles formula for sine: sin(x + y) = sin(x)cos(y) + cos(x)sin(y).
Given that sin(x) = 1/5 and cos(y) = 2/3, we need to find cos(x) and sin(y). Since x and y are in the first quadrant, sin(x) > 0 and cos(y) > 0.
To find cos(x), we can use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
Given sin(x) = 1/5, we can square both sides: (1/5)^2 + cos^2(x) = 1.
Simplifying: 1/25 + cos^2(x) = 1.
Subtracting 1/25 from both sides: cos^2(x) = 24/25.
Taking the square root of both sides (since cos(x) > 0): cos(x) = sqrt(24/25).
Since y is in the first quadrant and cos(y) = 2/3, we know sin^2(y) = 1 - cos^2(y).
Given cos(y) = 2/3, we can square both sides: (2/3)^2 + sin^2(y) = 1.
Simplifying: 4/9 + sin^2(y) = 1.
Subtracting 4/9 from both sides: sin^2(y) = 5/9.
Taking the square root of both sides (since sin(y) > 0): sin(y) = sqrt(5/9).
Now we have sin(x), cos(x), sin(y), cos(y), so we can substitute these values back into the formula sin(x + y) = sin(x)cos(y) + cos(x)sin(y).
sin(x + y) = (1/5)(2/3) + sqrt(24/25)(sqrt(5/9)).
Multiplying the fractions: sin(x + y) = 2/15 + (sqrt(120) / (sqrt(225)).
Simplifying: sin(x + y) = 2/15 + (sqrt(120)/15).
Therefore, sin(x + y) = (2 + sqrt(120))/15.
if sin a = 1/5
then cos a = sqrt(1 - 1/25)
= sqrt (24/25) = (2/5) sqrt 6
if cos b = 2/3
then sin b = sqrt ( 1 - 4/9)
= (1/3) sqrt 5
then use
sin (a+b) = sin a cos b + cos a sin b