To find the values of trigonometric functions given that sin(A) = 4/7, where π/2 < A < π, you can use the Pythagorean identity and the definitions of these trigonometric functions:
(a) cos(A):
You can use the Pythagorean identity, sin^2(A) + cos^2(A) = 1, to find cos(A):
cos^2(A) = 1 - sin^2(A)
cos^2(A) = 1 - (4/7)^2
cos^2(A) = 1 - 16/49
cos^2(A) = 33/49
cos(A) = ±√(33/49)
Since A is in the second quadrant (π/2 < A < π), cos(A) is negative:
cos(A) = -√(33/49)
(b) tan(A):
You can find tan(A) using the definition of tan(A) = sin(A) / cos(A):
tan(A) = (4/7) / (-√(33/49))
tan(A) = -(4/7) / √(33/49)
tan(A) = -(4/7) / (7/√33)
tan(A) = -4 / (7√33)
(c) csc(A):
Cosecant (csc) is the reciprocal of sine:
csc(A) = 1 / sin(A)
csc(A) = 1 / (4/7)
csc(A) = 7/4
(d) sec(A):
Secant (sec) is the reciprocal of cosine:
sec(A) = 1 / cos(A)
sec(A) = 1 / (-√(33/49))
sec(A) = -7/√33
(e) cot(A):
Cotangent (cot) is the reciprocal of tangent:
cot(A) = 1 / tan(A)
cot(A) = 1 / (-4 / (7√33))
cot(A) = -(7√33) / 4
So, the values are:
(a) cos(A) = -√(33/49)
(b) tan(A) = -4 / (7√33)
(c) csc(A) = 7/4
(d) sec(A) = -7/√33
(e) cot(A) = -(7√33) / 4