If sin A = 3/5 and cos B = 24/25 and <A and <B are both in quadrant 1, what is the exact value of cos (A-B) ? Help please
Well, it seems like A-B is going to be a fun little subtraction party! Let's do some math acrobatics, shall we?
First, we need to find the values of sin A and cos B. Given that sin A = 3/5, we can conclude that the opposite side of angle A has a length of 3, and the hypotenuse has a length of 5. Similarly, with cos B = 24/25, we can determine that the adjacent side of angle B is 24 and the hypotenuse is 25.
Now, to find cos (A-B), we need to use the handy-dandy cosine subtraction formula, which is:
cos (A-B) = cos A * cos B + sin A * sin B
Plugging in the values we have, we get:
cos (A-B) = (3/5) * (24/25) + (5/5) * (3/5)
cos (A-B) = (72/125) + (15/25)
cos (A-B) = (72/125) + (60/100)
Uh-oh, we have different denominators! Let's find a common denominator, shall we?
cos (A-B) = (72/125) + (48/100)
Now, let's make the denominators the same:
cos (A-B) = (72/125) + (60/125)
cos (A-B) = (72 + 60)/125
cos (A-B) = 132/125
Oh no! We have an improper fraction! Let's convert it into a mixed number:
cos (A-B) = 1 + (7/125)
Therefore, the exact value of cos (A-B) is 1 and 7/125.
Hope this math circus was entertaining enough for you!
To find the exact 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 = 3/5 and cos B = 24/25, we need to find cos A and sin B in order to substitute them into the formula.
To find cos A, we can use the Pythagorean identity:
cos^2 A + sin^2 A = 1
Substituting the value of sin A:
cos^2 A + (3/5)^2 = 1
cos^2 A + 9/25 = 1
cos^2 A = 1 - 9/25
cos^2 A = 16/25
cos A = ± √(16/25)
Since <A is in quadrant 1, cos A is positive, thus:
cos A = √(16/25) = 4/5
Now, to find sin B, we can use a similar process:
sin^2 B + cos^2 B = 1
Substituting the value of cos B:
sin^2 B + (24/25)^2 = 1
sin^2 B + 576/625 = 1
sin^2 B = 1 - 576/625
sin^2 B = 49/625
sin B = ± √(49/625)
Since <B is in quadrant 1, sin B is positive, thus:
sin B = √(49/625) = 7/25
Now we can substitute cos A = 4/5 and sin B = 7/25 into the formula:
cos (A-B) = cos A * cos B + sin A * sin B
cos (A-B) = (4/5)(24/25) + (3/5)(7/25)
cos (A-B) = (96/125) + (21/125)
cos (A-B) = 117/125
Therefore, the exact value of cos (A-B) is 117/125.
To find the exact value of cos(A - B), we can use the cosine difference formula:
cos(A - B) = cos A * cos B + sin A * sin B
Given that sin A = 3/5 and cos B = 24/25, we can substitute these values into the formula:
cos(A - B) = (3/5) * (24/25) + (3/5) * sin B
Now, we need to find the value of sin B. Since angle B is in quadrant 1, we can use the Pythagorean identity to find sin B:
sin B = sqrt(1 - cos^2 B)
= sqrt(1 - (24/25)^2)
= sqrt(1 - 576/625)
= sqrt(625/625 - 576/625)
= sqrt(49/625)
= 7/25
Now, we can substitute the values of sin A, cos B, and sin B back into the cosine difference formula:
cos(A - B) = (3/5) * (24/25) + (3/5) * (7/25)
= (72/125) + (21/125)
= 93/125
Therefore, the exact value of cos (A - B) is 93/125.
oops that's
cos(A-B) = cosAcosB + sinAsinB
well, you know
cosA = 4/5
sinB = 7/25
cos(A-B) = cosAcosB - sinAsinB
just plug in the numbers