If cos A =1/3 with A in QIV find cscA/2
Why did the angle go to a party in Quadrant IV? It wanted to get away from all the negative vibes in Quadrant III, of course! Now, let's calculate csc(A/2) and make sure it's having a good time too.
Since cos(A) = 1/3, we can determine that sin(A) = sqrt(1 - cos^2(A)) = sqrt(1 - (1/3)^2) = sqrt(1 - 1/9) = sqrt(8/9) = (2 * sqrt(2))/3.
To find csc(A/2), we need to calculate sin(A/2) first. Using the half-angle formula for sine, we have sin(A/2) = sqrt((1 - cos(A))/2) = sqrt((1 - 1/3)/2) = sqrt(2/6) = sqrt(1/3) = 1/sqrt(3).
Now, we can calculate csc(A/2) by taking the reciprocal of sin(A/2): csc(A/2) = 1/(1/sqrt(3)) = sqrt(3).
So, csc(A/2) is having a blast at the party and equals sqrt(3)!
First find sin (A/2)= +/-sqrt[(1/2)(1 - cos A)]
With A in the fourth quadrant, A/2 is in the second quadrant and sin A/2 is postive. Therefore
sin (A/2) = -sqrt[(1/2)(2/3)]
= sqrt(1/3)
csc (A/2) is 1/cos(A/2)= sqrt 3
check: A = 289.47 degrees
A/2 = 144.736 degrees
sin A/2 = 0.57735 = 1/(sqrt 3)
csc A/2 = sqrt 3
To find csc(A/2), we can first find the value of sin(A/2) using the half angle formula.
Given: cos A = 1/3 in Quadrant IV
Step 1: Find sin A using the Pythagorean identity.
Since cos A = 1/3, we can use the equation cos^2(A) + sin^2(A) = 1.
Substituting the value of cos A, we get (1/3)^2 + sin^2(A) = 1.
1/9 + sin^2(A) = 1.
sin^2(A) = 1 - 1/9.
sin^2(A) = 8/9.
Since we are in Quadrant IV, sin A should be positive.
Taking the square root, we get sin A = sqrt(8)/3.
Step 2: Use the half-angle formula to find sin(A/2).
The half-angle formula is given as sin(A/2) = sqrt((1 - cos A) / 2).
Substituting the value of cos A, we have:
sin(A/2) = sqrt((1 - (1/3)) / 2).
sin(A/2) = sqrt((2/3) / 2).
sin(A/2) = sqrt(1/3).
Step 3: Find csc(A/2) by taking the reciprocal of sin(A/2).
csc(A/2) = 1 / sin(A/2).
csc(A/2) = 1 / (sqrt(1/3)).
csc(A/2) = sqrt(3).
Therefore, csc(A/2) = sqrt(3).
To find csc(A/2) given that cos(A) = 1/3 with A in Quadrant IV, we first need to find the value of sin(A).
Since cos(A) = 1/3 in Quadrant IV, we can use the Pythagorean identity to find sin(A):
sin^2(A) = 1 - cos^2(A)
sin^2(A) = 1 - (1/3)^2
sin^2(A) = 1 - 1/9
sin^2(A) = 8/9
Now, we can find sin(A) by taking the square root of both sides of the equation:
sin(A) = √(8/9)
sin(A) = √8/√9
sin(A) = √8/3
Next, we need to find the value of csc(A/2) using the half-angle identity for cosecant:
csc(A/2) = 1/sin(A/2)
To find sin(A/2), we can use the half-angle formula for sine:
sin(A/2) = ±√((1 - cos(A))/2)
Since A is in Quadrant IV, sin(A/2) will be positive. Therefore, we take the positive square root:
sin(A/2) = √((1 - cos(A))/2)
sin(A/2) = √((1 - 1/3)/2)
sin(A/2) = √(2/3)
Finally, we can substitute sin(A/2) into the expression for csc(A/2):
csc(A/2) = 1/sin(A/2)
csc(A/2) = 1/(√(2/3))
csc(A/2) = √(3/2)/2
So, csc(A/2) = √(3/2)/2.