if sinA=-(12/13) and -90<A<0, find without using calcuator, the value of cot(90-A)
cot(90-A)= -2.40 approximately
use arcsinA or 1/sinA =-(12/13) to get the value of A..or use sine graph..to get the answer.
cotangent=cosx/sinx..
just input the value of A which is -67.38 to that of formula above with x as (90-A)...
It said "without a calculator"
All the work following is done in my head.
sinA = -12/13 = y/r
x^2 + y^2 = r^2
x^2 + (-12)^2 = 13^2
x^2 + 144 = 169
x^2 = 25
x = ± 5 , but A is in quad IV, so x = 5
so sinA = -12/13, cosA = +5/13
tanA = (-12/13) / (5/13) = -12/5
since cot(90-A) = tanA
cot(90-A) = -12/5 = -2.4
(exactly, not approximately)
To solve this problem without using a calculator, we can use the concept of complementary angles.
First, let's find the value of cos(A). We know that sin(A) = -(12/13), and we also know that sin^2(A) + cos^2(A) = 1. Plugging in the given value of sin(A), we can solve for cos(A):
(-12/13)^2 + cos^2(A) = 1
144/169 + cos^2(A) = 1
cos^2(A) = 1 - 144/169
cos^2(A) = 25/169
cos(A) = ±√(25/169)
cos(A) = ±(5/13)
Since -90° < A < 0°, A is in the 4th quadrant, where both sin and cos are negative. Therefore, cos(A) = -(5/13).
Now, let's find cot(90 - A). We know that cot(90 - A) = cos(A) / sin(90 - A). We already have the value of cos(A), so we need to find sin(90 - A).
Using the identity sin(90 - A) = cos(A), we can substitute cos(A) into the expression for cot(90 - A):
cot(90 - A) = cos(A) / sin(90 - A)
cot(90 - A) = -(5/13) / cos(A)
cot(90 - A) = -(5/13) / (-(5/13))
cot(90 - A) = 1
Therefore, without using a calculator, the value of cot(90 - A) is 1.