find sin(s+t), given that sin t=1/3, t in quadrant 2, and cos s = -2/5, s in quadrant 3
To find sin(s+t), we can use the trigonometric identity: sin(s+t) = sin s * cos t + cos s * sin t.
Given that sin t = 1/3 and t is in quadrant 2, we can determine that cos t is negative.
To find cos t, we can use the Pythagorean identity sin^2 t + cos^2 t = 1. Since sin t = 1/3, we have:
(1/3)^2 + cos^2 t = 1
1/9 + cos^2 t = 1
cos^2 t = 1 - 1/9
cos^2 t = 8/9
cos t = -√(8/9) (since cos t is negative in quadrant 2)
Given that cos s = -2/5 and s is in quadrant 3, we can determine that sin s is negative.
To find sin s, we can use the Pythagorean identity sin^2 s + cos^2 s = 1. Since cos s = -2/5, we have:
sin^2 s + (-2/5)^2 = 1
sin^2 s + 4/25 = 1
sin^2 s = 1 - 4/25
sin^2 s = 21/25
sin s = -√(21/25) (since sin s is negative in quadrant 3)
Now, we can substitute the values into the trigonometric identity:
sin(s+t) = sin s * cos t + cos s * sin t
sin(s+t) = (-√(21/25)) * (-√(8/9)) + (-2/5) * (1/3)
sin(s+t) = (√(21/25)) * (√(8/9)) + (-2/5) * (1/3)
sin(s+t) = (√(21 * 8) / √(25 * 9)) + (-2/5) * (1/3)
sin(s+t) = (√(168) / √(225)) + (-2/5) * (1/3)
sin(s+t) = (√(168) / 15) - 2/15
sin(s+t) = (√168 - 2) / 15
Therefore, sin(s+t) = (√168 - 2) / 15.
To find sin(s+t), we will use the trigonometric identity:
sin(s+t) = sin(s) * cos(t) + cos(s) * sin(t)
Given that sin(t) = 1/3 in quadrant 2, we can determine that the opposite side of the triangle is 1, and the hypotenuse is 3 (since sin(theta) = opposite/hypotenuse).
Next, given that cos(s) = -2/5 in quadrant 3, we can determine that the adjacent side of the triangle is -2 and the hypotenuse is 5 (since cos(theta) = adjacent/hypotenuse).
Now, we can calculate the opposite side of angle s using the Pythagorean theorem:
opposite side = sqrt(hypotenuse^2 - adjacent side^2)
= sqrt(5^2 - (-2)^2)
= sqrt(25 - 4)
= sqrt(21)
Now we have all the components we need to calculate sin(s+t):
sin(s+t) = sin(s) * cos(t) + cos(s) * sin(t)
= (sqrt(21)/5) * (1/3) + (-2/5) * (1/3)
= sqrt(21)/15 - 2/15
= (sqrt(21) - 2)/15
Therefore, sin(s+t) equals (sqrt(21) - 2)/15.