find the exact value of sin(u+v) given that sin u=3/5 and sin v= 15/17
To find the exact value of sin(u+v) given sin u = 3/5 and sin v = 15/17, we can use the trigonometric identity for the sine function:
sin(u+v) = sin u * cos v + cos u * sin v
First, we need to find the values of cos u and cos v. Since sin^2 u + cos^2 u = 1, we can solve for cos u:
cos^2 u = 1 - sin^2 u
cos^2 u = 1 - (3/5)^2
cos^2 u = 1 - 9/25
cos^2 u = 16/25
Taking the square root of both sides, we have:
cos u = ±4/5
Since sin u is positive (3/5), we can conclude that cos u is also positive (4/5).
Similarly, we can find the value of cos v:
cos^2 v = 1 - sin^2 v
cos^2 v = 1 - (15/17)^2
cos^2 v = 1 - 225/289
cos^2 v = 64/289
Taking the square root of both sides, we have:
cos v = ±8/17
Since sin v is positive (15/17), we can conclude that cos v is also positive (8/17).
Now that we have the values of sin u, cos u, sin v, and cos v, we can substitute them into our trigonometric identity to find the value of sin(u+v):
sin(u+v) = (3/5)(8/17) + (4/5)(15/17)
= 24/85 + 60/85
= 84/85
Therefore, the exact value of sin(u+v) is 84/85.
To find the exact value of sin(u+v) given that sin u = 3/5 and sin v = 15/17, we can use the sum formula for sine:
sin(u+v) = sin u * cos v + cos u * sin v
To find sin(u+v), we need to find the values of cos u and cos v. To do this, we can use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
Given that sin u = 3/5, we can find cos u as follows:
sin^2(u) + cos^2(u) = 1
(3/5)^2 + cos^2(u) = 1
9/25 + cos^2(u) = 1
cos^2(u) = 1 - 9/25
cos^2(u) = 25/25 - 9/25
cos^2(u) = 16/25
cos u = ± √(16/25)
cos u = ± 4/5
Since sin u is positive, cos u must also be positive. Therefore: cos u = 4/5.
Using the same logic, we can find cos v:
sin^2(v) + cos^2(v) = 1
(15/17)^2 + cos^2(v) = 1
225/289 + cos^2(v) = 1
cos^2(v) = 1 - 225/289
cos^2(v) = 289/289 - 225/289
cos^2(v) = 64/289
cos v = ± √(64/289)
cos v = ± 8/17
Since sin v is positive, cos v must also be positive. Therefore: cos v = 8/17.
Now we can substitute the values of sin u, sin v, cos u, and cos v into the sum formula for sine:
sin(u+v) = sin u * cos v + cos u * sin v
sin(u+v) = (3/5) * (8/17) + (4/5) * (15/17)
To simplify the expression, we can multiply the numerators together and multiply the denominators together:
sin(u+v) = (3 * 8) / (5 * 17) + (4 * 15) / (5 * 17)
sin(u+v) = 24/85 + 60/85
Finally, we can combine the fractions:
sin(u+v) = (24 + 60) / 85
sin(u+v) = 84/85
Therefore, the exact value of sin(u+v) is 84/85.
just drag out your good old trig identities:
sin^2 a + cos^2 a = 1
You have a 3-4-5 and a 8-15-17 triangle.
You could figure that out using the formula above.
Next:
sin(u+v) = sinu cosv + cosu sinv
sin(u+v) = 3/5 * 8/17 + 4/5 * 15/17
and I guess you can take it from there . . .