Find the exact value ofcos (u+v) given that sin u= 7/25 and cos v=−12/13. (Both u and v are in Quadrant II.)
Hmmm. I get
cosu*cosv - sinu*sinv
= (-24/25)(-12/13)-(7/25)(5/13)
= 288/325 - 35/325
= 253/325
since y>0 and x<0, draw your triangles, and you can see that
cos u = -24/25
sin v = 5/13
Now just plug them into your cos(u+v) formula.
Oh, I see you're playing around with trigonometry! Well, let's see what we can do here. Since both u and v are in Quadrant II, we know that sin u is positive and cos v is negative.
Now, we can use the Pythagorean identity sin²(u) + cos²(u) = 1 to find cos u. If sin u is 7/25, we can square it and subtract from 1 to find cos u. But hey, let me warn you, this math business can get a bit square, if you know what I mean!
So, sin²(u) = (7/25)² = 49/625. Now, to find cos u, we subtract that from 1: 1 - 49/625 = 576/625. Looks like we're getting there!
Now let's move on to cos v. You've mentioned that it's -12/13. Well, in Quadrant II, the cosine is negative, so we're on the right path.
To find the exact value of cos (u + v), we'll use the cosine addition formula, which states that cos (u + v) = cos u * cos v - sin u * sin v.
Now we can substitute the values we found:
cos (u + v) = (576/625) * (-12/13) - (7/25) * sin v.
But hey, looks like we're missing something here. You forgot to mention the value of sin v! Without it, I'm afraid I won't be able to provide you with the exact value of cos (u + v). Mind giving me the missing piece of this trigonometric puzzle?
To find the exact value of cos(u + v), we will use the cosine addition formula:
cos(u + v) = cos u * cos v - sin u * sin v
Given that sin u = 7/25 and cos v = -12/13, we can substitute the values into the formula:
cos(u + v) = cos u * cos v - sin u * sin v
cos(u + v) = (7/25) * (-12/13) - (7/25) * (sin v)
To find sin v, we can use the Pythagorean identity:
sin^2 v + cos^2 v = 1
Given that cos v = -12/13, we can solve for sin v:
sin^2 v + (-12/13)^2 = 1
sin^2 v + 144/169 = 1
sin^2 v = 1 - 144/169
sin^2 v = 169/169 - 144/169
sin^2 v = 25/169
sin v = √(25/169)
sin v = 5/13 (Since sin v is positive in Quadrant II)
Now we can substitute the values of sin u, cos v, and sin v into the equation for cos(u + v):
cos(u + v) = (7/25) * (-12/13) - (7/25) * (5/13)
cos(u + v) = -84/325 - 35/325
cos(u + v) = -119/325
Therefore, the exact value of cos(u + v) is -119/325.
so I got cos(u+v)=204/325
is this correct?