Given that cosA=4/5 and sinB=15/17 where A and B are acute angles,find the value of cos(A-B)
or, if you just recognize a few basic Pythagorean triples, like 3-4-5, 5-12-13, and 8-15-17 you could have skipped directly to the last step.
To find the value of cos(A - B), we can use the following trigonometric identity:
cos(A - B) = cosA * cosB + sinA * sinB
Given that cosA = 4/5 and sinB = 15/17, we need to find the values of cosB and sinA.
To find cosB, we can use the Pythagorean identity:
cos^2(B) + sin^2(B) = 1
Given that sinB = 15/17, we can substitute this value:
cos^2(B) + (15/17)^2 = 1
cos^2(B) + 225/289 = 1
cos^2(B) = 1 - 225/289
cos^2(B) = 289/289 - 225/289
cos^2(B) = 64/289
cosB = sqrt(64/289)
cosB = 8/17
To find sinA, we can use the Pythagorean identity as well:
cos^2(A) + sin^2(A) = 1
Given that cosA = 4/5, we can substitute this value:
(4/5)^2 + sin^2(A) = 1
16/25 + sin^2(A) = 1
sin^2(A) = 1 - 16/25
sin^2(A) = 25/25 - 16/25
sin^2(A) = 9/25
sinA = sqrt(9/25)
sinA = 3/5
Now, we can substitute the values of cosA, sinB, cosB, and sinA into the formula for cos(A - B):
cos(A - B) = cosA * cosB + sinA * sinB
cos(A - B) = (4/5) * (8/17) + (3/5) * (15/17)
cos(A - B) = (32/85) + (45/85)
cos(A - B) = 77/85
Therefore, the value of cos(A - B) is 77/85.
To find the value of cos(A-B), we can use the formula for cosine of the difference of two angles:
cos(A-B) = cosA * cosB + sinA * sinB
Given that cosA = 4/5 and sinB = 15/17, we still need to find the values of sinA and cosB.
To find sinA, we can use the Pythagorean identity: sin^2(A) + cos^2(A) = 1. Since we already know cosA, we can substitute it into the formula and solve for sinA:
sin^2(A) + (4/5)^2 = 1
sin^2(A) + 16/25 = 1
sin^2(A) = 1 - 16/25
sin^2(A) = 25/25 - 16/25
sin^2(A) = 9/25
sin(A) = √(9/25)
sin(A) = 3/5
Now we can substitute the values into the formula for cos(A-B):
cos(A-B) = (4/5) * cosB + (3/5) * (15/17)
To evaluate this expression, multiply the numerators and denominators:
cos(A-B) = (4/5) * cosB + (3/5) * (15/17)
cos(A-B) = (4 * cosB) / 5 + (3 * 15) / (5 * 17)
cos(A-B) = (4 * cosB) / 5 + 45 / 85
cos(A-B) = (4 * cosB) / 5 + (9/17)
The value of cos(A-B) cannot be determined without knowing the value of cosB.
sketch right-angled triangles that have the given information shown,
both in quadrant I
cosA = 4/5 = x/r --->x=4 and r=5
x^2 + y^2 = r^2
16 + y^2 = 25
y = √9 = 3
then sinA = 3/5
sinB=15/17 = y/r, so y=15, r = 17
x^2 + 15^2 = 17^2
x = √64 = 8
cosB = 8/17
You should know that
cos(A-B) = cosAcosB + sinAsinB
= ... , you know all the values
(4/5)(8/17) + 3/5(15/17) =