If sin x = -1/3 is in the 3rd quadrant and cos y = 2/5 is in the 4th quadrant, find the value of sec (x-y)
cos(x-y) = cosx cosy + sinx siny = (-√8/3)(2/5) + (-1/3)(-√21/5)
sin x = -1/3 is in the 3rd quadrant
then x = -1, r = 3 and
x^2 + y^2 = r^2
1 + y^2 = 9
y = √8 = -2√2
so we have cosx = -1/3, and sinx = -2√2/3
cos y = 2/5 is in the 4th quadrant
2^2 + y^2 = 25
y = -√21
so we have cosy = 2/5, and siny = -√21/5
cos(x-y) = cosxcosy + sinxsiny
= ... , you have those values, plug them in, then
sec(x-y) = 1/cos(x-y) , flip your fraction
Error!
In the first section , I interchanged sinx and cosx
go with ooblecks's numbers
To find the value of sec(x-y), we need to determine the values of x and y first.
Given that sin x = -1/3 is in the 3rd quadrant, we know that sine is negative in the 3rd quadrant. Since sine is opposite to the hypotenuse, we can imagine a right-angled triangle in the 3rd quadrant where the opposite side is -1 and the hypotenuse is 3. Using the Pythagorean theorem, we can find the value of the adjacent side:
adjacent side^2 = hypotenuse^2 - opposite side^2
adjacent side^2 = 3^2 - (-1)^2
adjacent side^2 = 9 - 1
adjacent side^2 = 8
adjacent side = √8 = 2√2
Therefore, in the 3rd quadrant, sin x = -1/3 corresponds to an angle with a sin ratio of -1/3, an opposite side length of -1, an adjacent side length of 2√2, and a hypotenuse length of 3.
Now, let's look at cos y = 2/5 in the 4th quadrant. In the 4th quadrant, cosine is positive. Similarly, we can imagine a right-angled triangle in the 4th quadrant where the adjacent side is 2 and the hypotenuse is 5. Using the Pythagorean theorem, we can find the value of the opposite side:
opposite side^2 = hypotenuse^2 - adjacent side^2
opposite side^2 = 5^2 - 2^2
opposite side^2 = 25 - 4
opposite side^2 = 21
opposite side = √21
Therefore, in the 4th quadrant, cos y = 2/5 corresponds to an angle with a cos ratio of 2/5, an adjacent side length of 2, an opposite side length of √21, and a hypotenuse length of 5.
Now, we have the values of sin x and cos y. We can combine these values using the double-angle identity for sec:
sec(x-y) = 1 / cos(x-y)
To evaluate cos(x-y), we need to understand the relationship between sin and cos in terms of addition and subtraction angles:
cos(x-y) = cos x * cos y + sin x * sin y
Plugging in the values, we get:
cos(x-y) = (2/5) * (2) + (-1/3) * (√21)
cos(x-y) = 4/5 - (√21)/3
Finally, substituting this value into the original equation for sec(x-y):
sec(x-y) = 1 / (4/5 - (√21)/3)
To simplify it further, we should multiply the numerator and denominator by the conjugate of the denominator:
sec(x-y) = (1 * (3(4/5 - (√21)/3))) / ((4/5 - (√21)/3) * (3(4/5 - (√21)/3)))
sec(x-y) = 3(4/5 - (√21)/3) / (12/15 - (√21)/5 - (√21)/5 + 21/9)
sec(x-y) = 3(4/5 - (√21)/3) / (12/15 - (2(√21))/5 + 21/9)
sec(x-y) = 3(4/5 - (√21)/3) / (36/45 - (10(√21))/15 + 35/15)
sec(x-y) = 3(4/5 - (√21)/3) / (71/45 - (10(√21))/15)
sec(x-y) = 3(4/5 - (√21)/3) / (141/90 - (20(√21))/30)
sec(x-y) = 3(4/5 - (√21)/3) / (157/90 - (20(√21))/30)
Now, we can simplify the expression further by multiplying both numerator and denominator by the reciprocal of the denominator:
sec(x-y) = (3(4/5 - (√21)/3) * (90/157 + (20(√21))/30)) / 1
sec(x-y) = (3 * (4(90/157) - (20(√21))/30) + (-√21)(90/157) - (20(√21))/3) / 1
sec(x-y) = (12(90/157) - (60(√21))/30 - (30(√21))/30) + (-√21)(90/157)) / 1
sec(x-y) = (12(90/157) - (90(√21))/30) + (-√21)(90/157)) / 1
sec(x-y) = (1080/157 - 90(√21)/30 - 90(√21)/157) / 1
sec(x-y) = (1080/157 - 3(√21) - 90(√21)/157) / 1
sec(x-y) = (1080/157 - 93(√21)/157) / 1
sec(x-y) = (1080 - 93(√21))/157
Therefore, the value of sec(x-y) is (1080 - 93(√21))/157.