If sin A = - 3/5 with A in QIV , find cos(A/2)
The terminal point is at (4,-3)
cosA = 4/5, so
2cos^2(A/2) - 1 = 4/5
and since 3π/2 < A < 2π,
3π/4 < A/2 < π
That is, A/2 is in QII, so it is negative.
To find cos(A/2) knowing sin A, you can use the half-angle identity for cosine.
The half-angle identity for cosine states that:
cos(A/2) = ± sqrt((1 + cos A) / 2)
To determine the sign, you need to determine the quadrant of angle A/2.
Given that A is in QIV (quadrant IV), we know that A lies between 180 degrees and 270 degrees (or between π and 3π/2 radians).
Since sin A = -3/5, we can find cos A using the Pythagorean identity:
cos A = ± sqrt(1 - sin^2 A)
= ± sqrt(1 - (-3/5)^2)
= ± sqrt(1 - 9/25)
= ± sqrt(16/25)
= ± 4/5
Since A is in QIV, we know that cos A is positive. Therefore, cos A = 4/5.
Now, substituting cos A into the half-angle identity for cosine:
cos(A/2) = sqrt((1 + cos A) / 2)
= sqrt((1 + 4/5) / 2)
= sqrt(9/10)
= 3/√10
So, cos(A/2) is equal to 3/√10.