given that sin alpha/2 = 2/3, and that alpha/2 is in quadrant II, determine the exact values of all trig functions for alpha.
let alpha/2 = A/2
sin A/2 = 2/3
then cos A/2 = √5/3
sin A = 2(sin A/2)(cos A/2) = 2(2/3)(√5/3) = 4√5/9
make a sketch,
opposite = 4√5, hypotenuse = 9
adj^2 + 80 = 81
adj = 1
sin alpha = 4√5/9
cos alpha= 1/9
tan alpha = 4√5
I am sure you can find the three reciprocal values
To determine the exact values of all trigonometric functions for alpha, we can use the given information and the trigonometric identity.
Given:
sin(alpha/2) = 2/3
alpha/2 is in quadrant II
We can start by utilizing the half-angle formula for sine:
sin(alpha/2) = ± sqrt((1 - cos(alpha))/2)
Since alpha/2 is in quadrant II, sine is positive, and we have:
2/3 = sqrt((1 - cos(alpha))/2)
Now, we can square both sides of the equation to eliminate the square root:
(2/3)^2 = (1 - cos(alpha))/2
4/9 = (1 - cos(alpha))/2
Next, we can solve for cos(alpha):
2 * (4/9) = 1 - cos(alpha)
8/9 = 1 - cos(alpha)
cos(alpha) = 1 - 8/9
cos(alpha) = 1/9
Now that we have the value of cos(alpha), we can use it to determine the values of the other trigonometric functions.
To find sin(alpha), we can use the Pythagorean identity:
sin^2(alpha) + cos^2(alpha) = 1
sin^2(alpha) + (1/9)^2 = 1
sin^2(alpha) + 1/81 = 1
sin^2(alpha) = 1 - 1/81
sin^2(alpha) = 80/81
Taking the square root of both sides:
sin(alpha) = ± sqrt(80/81)
Since alpha/2 is in quadrant II, sine is positive, so:
sin(alpha) = sqrt(80/81)
To find tan(alpha), we can divide sin(alpha) by cos(alpha):
tan(alpha) = sin(alpha) / cos(alpha)
tan(alpha) = sqrt(80/81) / (1/9)
tan(alpha) = sqrt(80/81) * (9/1)
tan(alpha) = 9 * sqrt(80/81)
Finally, we can determine the values of the remaining trigonometric functions:
cosec(alpha) = 1/sin(alpha)
sec(alpha) = 1/cos(alpha)
cot(alpha) = 1/tan(alpha)
Let's evaluate these values:
cosec(alpha) = 1/sqrt(80/81) = sqrt(81/80)
sec(alpha) = 1/(1/9) = 9/1 = 9
cot(alpha) = 1/(9 * sqrt(80/81)) = sqrt(81/80)/(9 * sqrt(80/81)) = 1/9
Therefore, the exact values of all trigonometric functions for alpha are as follows:
sin(alpha) = sqrt(80/81)
cos(alpha) = 1/9
tan(alpha) = 9 * sqrt(80/81)
cosec(alpha) = sqrt(81/80)
sec(alpha) = 9
cot(alpha) = 1/9