if sec x=the square root of 5/2 with angle x in quadrant IV and than y= -(1/3) with angle in quadrant II, find the value of sin (x-y)
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To find the value of sin(x - y), we need to find the values of sin(x) and sin(y) first.
Given that sec(x) = √(5/2) and angle x is in quadrant IV, we can use the reciprocal identity for secant to find cosine. Since sec(x) = 1/cos(x), we have:
cos(x) = 1 / sec(x) = 1 / √(5/2) = √(2/5)
Since angle x is in quadrant IV, cosine is positive, and we know that sine is positive in quadrant I. Therefore, sin(x) is positive.
Using the Pythagorean identity for sin^2(x) + cos^2(x) = 1, we have:
sin^2(x) + (√(2/5))^2 = 1
sin^2(x) + 2/5 = 1
sin^2(x) = 1 - 2/5
sin^2(x) = 3/5
sin(x) = ±√(3/5)
Since angle x is in quadrant IV, sin(x) is positive. Hence, sin(x) = √(3/5).
Now let's find the value of sin(y). Given that y = -(1/3) with angle y in quadrant II, we know that the sine function is positive in quadrant II. Hence, sin(y) = 1/3.
Finally, we can find the value of sin(x - y) by using the subtraction formula for sine:
sin(x - y) = sin(x) * cos(y) - cos(x) * sin(y)
Substituting the values we found earlier, we get:
sin(x - y) = (√(3/5)) * cos(y) - (√(2/5)) * (1/3)
To find the value of cos(y), we can use the Pythagorean identity sin^2(y) + cos^2(y) = 1:
(1/3)^2 + cos^2(y) = 1
1/9 + cos^2(y) = 1
cos^2(y) = 1 - 1/9
cos^2(y) = 8/9
cos(y) = ±√(8/9)
Since angle y is in quadrant II, cosine is negative. Hence, cos(y) = -√(8/9).
Now we can substitute these values into the equation for sin(x - y):
sin(x - y) = (√(3/5)) * (-√(8/9)) - (√(2/5)) * (1/3)
= -√(24/45) - (√(2/5)) * (1/3)
= -√(24/45) - √(2/15)
Therefore, the value of sin(x - y) is -√(24/45) - √(2/15).