Suppose sin A = 12/13 with 90º≤A≤180º. Suppose also that sin B = -7/25 with -90º≤B≤0º. Find tan (A – B).
sin^-1(12/13) = 67.38
180-67.39 = 112.62 = A
sin^-1(-7/25) = -16.26 = B
Tan ( A - B) = ???
I think that's how you approach it.. someone verify please?
I'd do it like this:
in QII if sinA = 12/13, cosA = -5/13
in QIV, if sinB = -7/25, cosB = 24/25
So,
tanA = -12/5
tanB = -7/24
tan(A-B) = ((-12/5)-(-7/24))/(1+(-12/5)(-7/24))
= (-253/120) / (204/120)
= -253/204
Thank you guys! :)That really helped! I understand now!
To find tan(A - B), we need to use the trigonometric identity for tangent:
tan(A - B) = (tan A - tan B) / (1 + tan A * tan B)
First, let's find the values of tan A and tan B.
To find tan A, we can use the given value of sin A and the Pythagorean Identity, sin^2 A + cos^2 A = 1. Since sin A = 12/13, we can find cos A as follows:
cos^2 A = 1 - sin^2 A
cos^2 A = 1 - (12/13)^2
cos^2 A = 1 - 144/169
cos^2 A = 25/169
Taking the square root of both sides:
cos A = √(25/169)
cos A = 5/13
Finally, we can calculate tan A:
tan A = sin A / cos A
tan A = (12/13) / (5/13)
tan A = 12/5
Similarly, to find tan B, we can use the given value of sin B and the Pythagorean Identity. Since sin B = -7/25, we can find cos B as follows:
cos^2 B = 1 - sin^2 B
cos^2 B = 1 - (-7/25)^2
cos^2 B = 1 - 49/625
cos^2 B = 576/625
Taking the square root of both sides:
cos B = √(576/625)
cos B = 24/25
Finally, we can calculate tan B:
tan B = sin B / cos B
tan B = (-7/25) / (24/25)
tan B = -7/24
Now we have the values of tan A and tan B. We can substitute them in the formula for tan(A - B):
tan(A - B) = (tan A - tan B) / (1 + tan A * tan B)
tan(A - B) = (12/5 - (-7/24)) / (1 + (12/5) * (-7/24))
tan(A - B) = (12/5 + 7/24) / (1 - (12/5) * (7/24))
Now we can compute the numerator and denominator separately:
Numerator:
12/5 + 7/24 = (288/120 + 35/120) / (1 - (84/120))
= (323/120) / (36/120)
Denominator:
1 - (12/5) * (7/24) = 1 - (84/120)
= 1 - 7/10
= 3/10
Finally, we can divide the numerator by the denominator:
tan(A - B) = (323/120) / (3/10)
= (323/120) * (10/3)
= 107/12
Therefore, tan(A - B) = 107/12.