Find the exact value of the trigonometric function given that sin u = 5/13 and cos v = -3/5. (Both u and v are in Quadrant II.)
Find csc(u-v).
First of all, I drew the triangles of u and v. Also, I know the formula of sin(u-v) is sin u * cos v - cos u * sin v. However, I am confused of what you do when you have csc(u-v). Would you do csc u * sec v - sec u * csc v?
Thanks.
no, that is wishful thinking.
csc(u-v) = 1/sin(u-v)
so work out the sin(u-v) according to your formula, the take the reciprocal of your final fraction.
Okay, I worked sin(u-v) out and I got 33/65. Would I then flip it over and get 65/33 as my final answer?
yes, that is also what I got
(and I checked it by finding the actual angles)
Okay thanks for your help!
To find csc(u-v), we need to use the reciprocal trigonometric functions.
First, let's find the values of sin u, cos u, and sin v.
Since sin u = 5/13, we know that the opposite side of angle u is 5, and the hypotenuse is 13. By using the Pythagorean theorem, we can find the adjacent side of angle u:
cos u = sqrt(1 - sin^2(u)) = sqrt(1 - (5/13)^2) = sqrt(1 - 25/169) = sqrt(144/169) = 12/13
Similarly, for angle v with cos v = -3/5, we know that the adjacent side of angle v is -3, and the hypotenuse is 5. Using the Pythagorean theorem, we can find the opposite side of angle v:
sin v = sqrt(1 - cos^2(v)) = sqrt(1 - (-3/5)^2)) = sqrt(1 - 9/25) = sqrt(16/25) = 4/5
Now we have sin u, cos u, sin v, and cos v. To find csc(u-v), we can use the reciprocal functions:
csc(u-v) = 1/(sin(u-v))
Now, we need to use the formula for sin(u-v):
sin(u-v) = sin u * cos v - cos u * sin v
Plugging in the values we have, we get:
sin(u-v) = sin u * cos v - cos u * sin v
= (5/13) * (-3/5) - (12/13) * (4/5)
= -15/65 - 48/65
= -63/65
Now, we can substitute this value back into the reciprocal function:
csc(u-v) = 1/(sin(u-v))
= 1/(-63/65)
= -65/63
Therefore, csc(u-v) = -65/63.