A and B are positive acute angles. if sin A=4/5 and cos B=8/17 find the value of tan (A-B)
is the answer =43/100
cosA = 3/5
sinB = 15/17
tanA = 4/3
tanB = 15/8
tan(A-B) = (tanA-tanB)/(1+tanAtanB)
= (4/3 - 15/8)/(1+(4/3)(15/8))
= (-13/24)/(7/2)
= -13/84
thanks
To find the value of tan(A-B), we can use the trigonometric identity:
tan(A-B) = (tan A - tan B) / (1 + tan A * tan B)
First, let's find the values of tan A and tan B using the given information.
Since sin A = 4/5, we can use the definition of sine to find the value of the opposite side and hypotenuse. Let's assume a right-angled triangle with angle A:
sin A = opposite / hypotenuse
4/5 = opposite / hypotenuse
Let's assume the opposite side is 4x and the hypotenuse is 5x (where x is a positive constant):
4/5 = 4x / 5x
1 = x
So, the opposite side is 4, and the hypotenuse is 5.
Now, let's find the value of tan A:
tan A = opposite / adjacent
tan A = 4 / 3
Similarly, we can find the value of tan B using the given information:
cos B = 8/17
adjacent / hypotenuse = 8/17
Let's assume the adjacent side is 8x and the hypotenuse is 17x (where x is a positive constant):
8/17 = 8x / 17x
1 = x
So, the adjacent side is 8, and the hypotenuse is 17.
Now, let's find the value of tan B:
tan B = opposite / adjacent
tan B = (17-8) / 8
tan B = 9 / 8
Finally, we can substitute the values of tan A and tan B into the formula for tan(A-B):
tan(A-B) = (tan A - tan B) / (1 + tan A * tan B)
tan(A-B) = (4/3 - 9/8) / (1 + (4/3)*(9/8))
tan(A-B) = (32/24 - 27/24) / (1 + (12/24))
tan(A-B) = (5/24) / (36/24)
tan(A-B) = 5/36
Therefore, the value of tan (A-B) is 5/36, not 43/100.
To find the value of tan (A-B), we'll first need to determine the values of both A and B.
Given:
sin A = 4/5
cos B = 8/17
Since sin A = opposite/hypotenuse, let's assume a right-angled triangle ABC, where angle A is the acute angle opposite to side BC. In this triangle, let AB be the side opposite A, AC be the hypotenuse, and BC be the side adjacent to the right angle.
So, let AB = 4 and AC = 5 in this triangle.
Using the Pythagorean theorem, we can find BC:
BC^2 = AC^2 - AB^2
BC^2 = 5^2 - 4^2
BC^2 = 25 - 16
BC^2 = 9
BC = 3
Now, let's consider angle B. Since cos B = adjacent/hypotenuse, we can assume another right-angled triangle DEF, where angle B is the acute angle adjacent to side EF. In this triangle, let DE be the side adjacent to B, EF be the hypotenuse, and DF be the side opposite B.
So, let DE = 8 and EF = 17 in this triangle.
Using the Pythagorean theorem, we can find DF:
DF^2 = EF^2 - DE^2
DF^2 = 17^2 - 8^2
DF^2 = 289 - 64
DF^2 = 225
DF = 15
Now, we have the values of AB, BC, DE, and DF, and we can use these values to determine the value of tan (A-B).
tan (A-B) can be expressed as (tan A - tan B) / (1 + tan A * tan B).
We know that tan A = AB/BC and tan B = DE/DF
Substituting the values we have:
tan A = 4/3
tan B = 8/15
Now, let's calculate tan (A - B):
tan (A - B) = (tan A - tan B) / (1 + tan A * tan B)
= (4/3 - 8/15) / (1 + (4/3)(8/15))
= (20/15 - 8/15) / (1 + (32/45))
= 12/15 / (1 + 32/45)
= 12/15 / (77/45)
= 12/15 * (45/77)
= 540/1155
= 12/26
= 6/13
Therefore, the value of tan (A-B) is 6/13, not 43/100.