Given Tan(A) = 5 in Quadrant III and Sin(B) = ⅔ in Quadrant II, find Cos(A-B).
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To find the value of Cos(A-B), we can use the trigonometric identity:
Cos(A - B) = Cos(A) * Cos(B) + Sin(A) * Sin(B)
We are given that Tan(A) = 5 and Sin(B) = ⅔. Let's start by finding the values of Cos(A) and Sin(A):
Since Tan(A) = opposite/adjacent, we can construct a right triangle in Quadrant III where the opposite side is 5 and the adjacent side is -1 (since it lies in Quadrant III). By using the Pythagorean theorem, we can find the hypotenuse:
Hypotenuse^2 = opposite^2 + adjacent^2
Hypotenuse^2 = 5^2 + (-1)^2
Hypotenuse^2 = 26
Hypotenuse = √26
Now, since we know the values of the opposite side (5) and the hypotenuse (√26), we can find the value of Sin(A):
Sin(A) = opposite/hypotenuse
Sin(A) = 5/√26
Next, let's find the value of Cos(B):
By using the given Sin(B) = ⅔, we can construct a right triangle in Quadrant II where the opposite side is 2 and the hypotenuse is 3. By using the Pythagorean theorem, we can find the adjacent side:
Adjacent^2 = hypotenuse^2 - opposite^2
Adjacent^2 = 3^2 - 2^2
Adjacent^2 = 5
Adjacent = √5
Now, since we know the values of the adjacent side (√5) and the hypotenuse (3), we can find the value of Cos(B):
Cos(B) = adjacent/hypotenuse
Cos(B) = √5/3
Finally, we can substitute the values of Cos(A), Sin(A), Cos(B), and Sin(B) into the trigonometric identity:
Cos(A - B) = Cos(A) * Cos(B) + Sin(A) * Sin(B)
Cos(A - B) = (√26) * (√5/3) + (5/√26) * (⅔)
To simplify, multiply the fractions:
Cos(A - B) = (√(26*5)/3) + (5/(3√26)) * (2/3)
Simplifying further:
Cos(A - B) = (√130/3) + (10/(9√26))
Therefore, the value of Cos(A-B) is (√130/3) + (10/(9√26)).