Angle x is in the second quadrant and angle y is in the first quadrant such that sinx=5/13 and cosy=3/5.
a) Determine an exact value for cosx
b) Determine an exact value for siny
a) Well, if sin(x) = 5/13 and x is in the second quadrant, we know that sin(x) > 0 and cos(x) < 0. Now, let's use the Pythagorean identity to find the exact value of cos(x):
sin²(x) + cos²(x) = 1
(5/13)² + cos²(x) = 1
25/169 + cos²(x) = 1
cos²(x) = 1 - 25/169
cos²(x) = 144/169
Since cos(x) < 0 (because x is in the second quadrant), we know that:
cos(x) = -√(144/169)
cos(x) = -12/13
Therefore, the exact value of cos(x) is -12/13.
b) Now, we're given that cos(y) = 3/5 and y is in the first quadrant. In the first quadrant, both sine and cosine are positive. So, let's use the Pythagorean identity again to find the exact value of sin(y):
sin²(y) + cos²(y) = 1
sin²(y) + (3/5)² = 1
sin²(y) + 9/25 = 1
sin²(y) = 1 - 9/25
sin²(y) = 16/25
Since sin(y) > 0 (because y is in the first quadrant), we know that:
sin(y) = √(16/25)
sin(y) = 4/5
Therefore, the exact value of sin(y) is 4/5.
To determine the exact value for cos(x), we can use the Pythagorean identity:
sin^2(x) + cos^2(x) = 1
Given that sin(x) = 5/13, we can substitute the value in the equation:
(5/13)^2 + cos^2(x) = 1
25/169 + cos^2(x) = 1
Now, subtract 25/169 from both sides:
cos^2(x) = 1 - 25/169
cos^2(x) = 144/169
Taking the square root of both sides, we get:
cos(x) = ±(12/13)
Since angle x is in the second quadrant, the cosine of angle x is negative. Therefore, cos(x) = -12/13.
To determine the exact value for sin(y), we can also use the Pythagorean identity:
sin^2(y) + cos^2(y) = 1
Given that cos(y) = 3/5, we can substitute the value in the equation:
sin^2(y) + (3/5)^2 = 1
sin^2(y) + 9/25 = 1
Now, subtract 9/25 from both sides:
sin^2(y) = 1 - 9/25
sin^2(y) = 16/25
Taking the square root of both sides, we get:
sin(y) = ±(4/5)
Since angle y is in the first quadrant, the sine of angle y is positive. Therefore, sin(y) = 4/5.
To determine the exact value for cos(x), we need to use the given information about sin(x) and cos(y).
Since x is in the second quadrant, the cosine function in this quadrant is negative. Therefore, we have:
cos(x) = -√(1 - sin^2(x))
Given that sin(x) = 5/13, we can substitute this into the equation:
cos(x) = -√(1 - (5/13)^2)
Now, let's simplify this:
cos(x) = -√(1 - 25/169)
cos(x) = -√(144/169)
Since x is in the second quadrant and cosine is negative in this quadrant, we take the negative root:
cos(x) = -12/13
Therefore, an exact value for cos(x) is -12/13.
Next, to determine the exact value for sin(y), we can use the given information about cos(y).
Since y is in the first quadrant, both sine and cosine functions are positive. Therefore, we have:
sin(y) = √(1 - cos^2(y))
Given that cos(y) = 3/5, we can substitute this into the equation:
sin(y) = √(1 - (3/5)^2)
Now, let's simplify this:
sin(y) = √(1 - 9/25)
sin(y) = √(16/25)
Since y is in the first quadrant and sine is positive in this quadrant, we take the positive root:
sin(y) = 4/5
Therefore, an exact value for sin(y) is 4/5.