Suppose that sin x = 1/5 and cos y = 2/3, and x and y are each angles in Quadrant 1. Determine sin(x+y).
make triangles for each case , then use Pythagoras
if sinx = 1/5, then cosx = √24/5
if cosy = 2/3 , the siny = √5/3
sin(x+y) = sinxcosy + cosxsiny
= (1/5)(2/3) + (√24/5)(√5/3)
= (2 + √120)/15
To find sin(x+y), we need to use a trigonometric identity called the sum of angles formula. It states that sin(x+y) can be written as sin(x)cos(y) + cos(x)sin(y).
Given sin(x) = 1/5 and cos(y) = 2/3, we need to find cos(x) and sin(y).
To find cos(x), we'll use the Pythagorean identity which states that sin^2(x) + cos^2(x) = 1. We know sin(x) = 1/5, so we can substitute and solve for cos(x).
(1/5)^2 + cos^2(x) = 1
1/25 + cos^2(x) = 1
cos^2(x) = 1 - 1/25
cos^2(x) = 24/25
cos(x) = √(24/25) = √24/5
Next, to find sin(y), we'll use the same Pythagorean identity. Given cos(y) = 2/3, we can substitute and solve for sin(y).
sin^2(y) + (2/3)^2 = 1
sin^2(y) + 4/9 = 1
sin^2(y) = 1 - 4/9
sin^2(y) = 5/9
sin(y) = √(5/9) = √5/3
Now we have sin(x), cos(x), sin(y), and cos(y). We can substitute these values into the sum of angles formula to find sin(x+y).
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
= (1/5)(2/3) + (√24/5)(√5/3)
= 2/15 + (√24/5)(√5/3)
= 2/15 + (√(24*5)/(5*3))
= 2/15 + (√120/15)
= 2/15 + (√(2^3 * 5)/(3*5))
= 2/15 + (√(2^3 * 5)/15)
= 2/15 + (2√5/15)
= (2 + 2√5)/15
Therefore, sin(x+y) = (2 + 2√5)/15.