if sinA=-(12/13) and -90<A<0, find without using calcuator, the value of cot(90-A)
Just draw your 5-12-13 triangle in standard position
cot(90-A) = tan(A) = y/x = -12/5
To find the value of cot(90-A), we need to determine the value of tan(90-A) first, and then take its reciprocal to get cot(90-A).
We are given that sinA = -(12/13), and -90 < A < 0. Since sinA is negative, we know that A lies in the third quadrant. In the third quadrant, the value of sin is negative, while the value of cos is positive.
Using the cofunction identity, sin(90-A) = cosA, we can find the value of cosA. Since sinA = -(12/13), we can determine the value of cosA using the Pythagorean identity sin^2A + cos^2A = 1:
(12/13)^2 + cos^2A = 1
144/169 + cos^2A = 1
cos^2A = 25/169
cosA = ±(5/13)
Since A lies in the third quadrant where cos is positive, we can conclude that cosA = 5/13.
Now we can find the value of tan(90-A) by using the identity tan(90-A) = sinA / cosA:
tan(90-A) = sinA / cosA
tan(90-A) = -(12/13) / (5/13)
tan(90-A) = -12/5
Finally, we can find the value of cot(90-A) by taking the reciprocal of tan(90-A):
cot(90-A) = 1 / tan(90-A)
cot(90-A) = 1 / (-12/5)
cot(90-A) = -5/12
Therefore, cot(90-A) = -5/12.