PLEASE HELP!!
sin(theta)= 1/3
pi/2 < theta < pi
Find:
cos(theta)= ______ ?
tan(theta)= ______ ?
cot(theta)= ______ ?
sec(theta)= ______ ?
csc(theta)= ______ ?
draw it !
side on x axis = sqrt (9-1) = sqrt 8 =2 sqrt 2
so cos = (2/3)sqrt 2
tan = 1/[2 sqrt 2] = (sqrt 2)/4
etc
So right usheid go sien
To find the values of cos(theta), tan(theta), cot(theta), sec(theta), and csc(theta) when sin(theta) = 1/3 and pi/2 < theta < pi, we can make use of the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1.
We are given that sin(theta) = 1/3. Substituting this value into the Pythagorean identity, we can solve for cos(theta).
(1/3)^2 + cos^2(theta) = 1
1/9 + cos^2(theta) = 1
cos^2(theta) = 1 - 1/9
cos^2(theta) = 8/9.
Taking the square root of both sides, we find:
cos(theta) = √(8/9).
To find the values of tan(theta), cot(theta), sec(theta), and csc(theta), we can make use of the following trigonometric identities:
tan(theta) = sin(theta) / cos(theta)
cot(theta) = 1 / tan(theta)
sec(theta) = 1 / cos(theta)
csc(theta) = 1 / sin(theta).
Plugging in the values of sin(theta) = 1/3 and cos(theta) = √(8/9), we can find the values of the remaining trigonometric functions:
tan(theta) = (1/3) / (√(8/9))
cot(theta) = 1 / ((1/3) / (√(8/9)))
sec(theta) = 1 / (√(8/9))
csc(theta) = 1 / (1/3).
Simplifying these expressions will give you the final values.