cosA = 1/8 with A in QIV, find cscA/2

make your sketch of the Pythagorean triangle in quadrant IV

since cos Ø = adjacent/hypotenuse = x/r
x = 1, r = 8 and x^2 + y^2 = r^2
1 + y^2 = 64
y = ±√63 , but in IV y = -√63
so sinA = -√63/8

we also know cos A = 1 - 2sin^2 (A/2)
2sin^2 (A/2) = 1 - 1/8 = 7/8
sin^2 (A/2) = 7/16
sin (A/2) = ±√7/4 , but if A is in IV, then A/2 is in II and sinA = √7/4

so cscA = 4/√7 or 4√7/7

To find csc(A/2), we first need to identify the value of sin(A/2).

Given that cos(A) = 1/8 in Quadrant IV, we can use the Pythagorean identity to find sin(A):

sin²(A) + cos²(A) = 1

sin²(A) + (1/8)² = 1

sin²(A) + 1/64 = 1

sin²(A) = 1 - 1/64

sin²(A) = 63/64

Taking the square root of both sides, we get:

sin(A) = ± √(63/64)

Since A is in Quadrant IV, sin(A) is positive:

sin(A) = √(63/64)

To find sin(A/2), we can use the half-angle identity:

sin(A/2) = ± √((1 - cos(A))/2)

Plugging in the value of cos(A) and simplifying, we get:

sin(A/2) = ± √((1 - 1/8)/2)

sin(A/2) = ± √((7/8)/2)

sin(A/2) = ± √(7/16)

Since A is in Quadrant IV, sin(A/2) is negative:

sin(A/2) = - √(7/16)

Now, to find csc(A/2), we can use the reciprocal identity:

csc(A/2) = 1/sin(A/2)

csc(A/2) = 1/(- √(7/16))

csc(A/2) = -1/√(7/16)

To rationalize the denominator, we can multiply the numerator and denominator by √16:

csc(A/2) = (-1/√(7/16)) * (√16/√16)

csc(A/2) = -√16 / √(7 × 16)

csc(A/2) = -4√2 / √112

Finally, we can simplify the expression by rationalizing the denominator:

csc(A/2) = -4√2 / (4√7)

csc(A/2) = -√2 / √7

Thus, csc(A/2) = -√2 / √7

To find csc(A/2), we first need to determine the value of sin(A/2).

Given that cos(A) = 1/8 and A is in the fourth quadrant (QIV), we can use the Pythagorean identity sin^2(A) + cos^2(A) = 1 to find sin(A).

Since A is in QIV, both sin(A) and cos(A) are negative. We can represent them as sin(A) = -sqrt(1 - cos^2(A)) and cos(A) = -1/8.

Plugging the values into the Pythagorean identity, we get:
sin^2(A) + (-1/8)^2 = 1
sin^2(A) + 1/64 = 1
sin^2(A) = 1 - 1/64
sin^2(A) = 63/64

Taking the square root of both sides gives us:
sin(A) = sqrt(63/64)
sin(A) = sqrt(63)/8

Now, let's find sin(A/2) using the half-angle formula:
sin(A/2) = sqrt((1 - cos(A))/2)
sin(A/2) = sqrt((1 - (-1/8))/2)
sin(A/2) = sqrt((1 + 1/8)/2)
sin(A/2) = sqrt(9/8)/2
sin(A/2) = sqrt(9)/2sqrt(8)
sin(A/2) = 3/2sqrt(8)

Finally, let's find csc(A/2) (cosecant of A/2):
csc(A/2) = 1/sin(A/2)
csc(A/2) = 1 / (3/2sqrt(8))
csc(A/2) = 2sqrt(8) / 3

Therefore, csc(A/2) = 2sqrt(8) / 3.