Assume that A and B are in standard position and that sin A = 1/2, cos A >0, tan B = 3/4, and sinB<0
Cos (A-B)
from sin A = 1/2, cos A >0, we know that A has to be in quadrant I, so cosA = √3/2
from tan B = 3/4, and sinB<0, we know that B must be in quadrante III, so sinB=-3/5 and cosB = -4/5
Cos(A-B) = cosAcosB + sinAsinB
= √3/2(-4/5) + 1/2(-3/5)
= (-4√3 - 3)/10
To find the value of cos(A-B), we can use the following formula:
cos(A-B) = cos A * cos B + sin A * sin B
Given that sin A = 1/2, cos A > 0, tan B = 3/4, and sin B < 0, we can find the values of sin A, cos A, sin B, and cos B.
Since sin A = 1/2, we know that the opposite side of angle A is 1 and the hypotenuse is 2.
Using the Pythagorean theorem, we can find the adjacent side of angle A:
cos^2 A = 2^2 - 1^2 = 3
cos A = √3
Since tan B = 3/4, we know that the opposite side of angle B is 3 and the adjacent side is 4.
Using the Pythagorean theorem, we can find the hypotenuse of angle B:
hypotenuse^2 = 4^2 + 3^2 = 25
hypotenuse = √25 = 5
Using the values we found, we can calculate sin B:
sin B = opposite/hypotenuse = -3/5
Now, let's substitute the values into the formula:
cos(A-B) = cos A * cos B + sin A * sin B
cos(A-B) = (√3)(4/5) + (1/2)(-3/5)
cos(A-B) = (4√3/5) + (-3/10)
cos(A-B) = (8√3 - 3)/10
Therefore, cos(A-B) = (8√3 - 3)/10.
To find the value of cos(A - B), we can use the trigonometric identity for the cosine of the difference of two angles:
cos(A - B) = cos A * cos B + sin A * sin B
From the given information, we know that sin A = 1/2, cos A > 0, tan B = 3/4, and sin B < 0.
Since sin A = 1/2 and cos A > 0, we can use the Pythagorean identity to find cos A:
cos^2 A = 1 - sin^2 A
cos^2 A = 1 - (1/2)^2
cos^2 A = 1 - 1/4
cos^2 A = 3/4
Since cos A > 0, we take the positive square root:
cos A = sqrt(3)/2
Now, let's find sin B. We know that tan B = 3/4, which means that opposite/adjacent = 3/4. Using the Pythagorean theorem, we can find the hypotenuse:
opposite = 3x
adjacent = 4x
hypotenuse = sqrt((3x)^2 + (4x)^2) = sqrt(9x^2 + 16x^2) = sqrt(25x^2) = 5x
Sin B is equal to opposite/hypotenuse, which is:
sin B = opposite/hypotenuse = 3x/5x = 3/5
Since sin B < 0, we know that B lies in the fourth quadrant.
Now that we have the values of cos A (sqrt(3)/2), sin A (1/2), cos B (adjacent/hypotenuse), and sin B (3/5), we can substitute them into the formula for cos(A - B):
cos(A - B) = cos A * cos B + sin A * sin B
cos(A - B) = (sqrt(3)/2) * (4/5) + (1/2) * (-3/5)
cos(A - B) = (4sqrt(3)/10) - (3/10)
cos(A - B) = (4sqrt(3) - 3)/10
Therefore, cos(A - B) is equal to (4sqrt(3) - 3)/10.