use a double-angle formula to find the exact value of cos2u when sinu=5/13 , where pi/2<u<pi
So far I went
cos2u=1-sin^2
=1-2(25/169)
=1-(50/169)
Correct answer but you also can write :
cos ( 2 u ) = cos ^ 2 ( u ) - sin ^ 2 u
In this case :
sin ( u ) = 5 / 13
sin ^ 2 ( u ) = ( 5 / 13 ) ^ 2 = 25 / 169
cos ^ 2 ( u ) = 1 - sin ^ 2 ( u ) =
1 - 25 / 169 = 169 / 169 - 25 / 169 = 144 / 169
cos ( 2 u ) = cos ^ 2 ( u ) - sin ^ 2 u =
144 / 169 - 25 / 169 = 119 / 169
correct. How can you be stuck there? Surely by the time you have learned trigonometry, you know how to work with fractions ... ?
hint: 1 = 169/169
OH my god I feel very dumb because 1/1 is 169/169 Thank you !
To find the exact value of cos(2u), we can use the double-angle formula for cosine, which states:
cos(2u) = cos^2(u) - sin^2(u)
Given that sin(u) = 5/13, we can use this information to find cos(u).
To find cos(u), we can use the Pythagorean identity:
sin^2(u) + cos^2(u) = 1
Plugging in the value of sin(u):
(5/13)^2 + cos^2(u) = 1
25/169 + cos^2(u) = 1
Solving for cos^2(u):
cos^2(u) = 1 - 25/169
cos^2(u) = 144/169
Now, we can substitute this value into the double-angle formula for cosine:
cos(2u) = cos^2(u) - sin^2(u)
cos(2u) = (144/169) - (25/169)
cos(2u) = 119/169
So, the exact value of cos(2u) when sin(u) = 5/13 is 119/169.