Question 978511: if sin(x)=1/3 and sec(y)=5/4, where 0 < x < pi/2 and 0 < y < pi/2, evaluate the expression cos (x-y), these < have the line under them so like equal to as well
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! sin(x) = 1/3
sec(y) = 5/4.
cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
this is found by looking up trig identities, of which cos(x-y) or cos(a-b) is one of them.
to solve, you need sin and cos of x and you need sin and cos of y.
sin(x) = 1/3.
1 = opposite
3 = hypotenuse.
by pythagorus:
o^2 + a^2 = h^2
this is short for opposite squared plus acjacent suared = hypotenuse squared.
this becomes:
1^2 + a^2 = 3^2 which becomes:
1 + a^2 = 9 which becomes:
a^2 = 8 which becomes:
a = sqrt(8).
you have:
sin(x) = 1/3
cos(x) = sqrt(8)/3
sec(y) = 5/4
since sec = 1/cos, this becomes:
1/cos(y) = 5/4 which becomes:
cos(y) = 4/5
by pythagorus:
o^2 + a^2 = h^2 which becomes:
o^2 + 4^2 = 5^2 which becomes:
o^2 + 16 = 25 which becomes:
o^2 = 25 - 16 = 9 which becomes:
o = 3
you have:
sin(x) = 1/3
cos(x) = sqrt(8)/3
sin(y) = 3/5
cos(y) = 4/5
you can now solve the identity of cos(x-y) = cos(x)cos(y) + sin(x)sin(y), which becomes:
cos(x-y) = sqrt(8)/3 * 4/5 + 1/3 * 3/5 which becomes:
cos(x-y) = .954247
that's your solution.
you can confirm by solving for each of the angles separately.
sin(x) = 1/3 gets you x = 19.471221
sin(y) = 3/5 gets you y = 36.869898
(x-y) gets you 19.471221 - 36.869898 = -17.398677 degrees.
cos(x-y) = cos(-17.398677) = .954245
answers are the same either way so it looks like you did good.
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