angle x is in the second quadrant and angel y is in the first quadrant and angle y is in the first quardrant such that sin x=5/13 and cos y = 3/5. Determine and exact value for cos x.
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if sinx = 5/13, and x is in quad II
then cosx = -12/13 (from the 5,12,13 triangle)
To determine the exact value of cos(x), we need to consider the given information about angle x and angle y.
First, let's draw a rough diagram to visualize the situation:
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Q2|Q1
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Q3|Q4
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Angle x is in the second quadrant, so it will have a negative value for cos(x) since cosine is positive only in the fourth quadrant.
Angle y is in the first quadrant, so both sine and cosine values are positive.
Now, let's use the Pythagorean identity to determine the missing value of sin(y):
sin^2(y) + cos^2(y) = 1
Given that cos(y) = 3/5:
sin^2(y) + (3/5)^2 = 1
sin^2(y) + 9/25 = 1
sin^2(y) = 1 - 9/25
sin^2(y) = 16/25
sin(y) = √(16/25)
sin(y) = 4/5
Since sin(x) = 5/13 and x is in the second quadrant, we can use the Pythagorean identity again to determine the missing value of cos(x):
sin^2(x) + cos^2(x) = 1
Given that sin(x) = 5/13:
(5/13)^2 + cos^2(x) = 1
25/169 + cos^2(x) = 1
cos^2(x) = 1 - 25/169
cos^2(x) = 144/169
cos(x) = ± √(144/169)
Since angle x is in the second quadrant, the cosine value will be negative:
cos(x) = -√(144/169)
cos(x) = -12/13
Therefore, the exact value of cos(x) is -12/13.