if sec theta =13/12 then find the value of sin theta and cot theta
To find the values of sin theta and cot theta given sec theta, we can use the following trigonometric identity:
sec^2 theta = 1 + tan^2 theta
Let's substitute the given value of sec theta into this equation:
(13/12)^2 = 1 + tan^2 theta
Simplifying this equation, we get:
169/144 = 1 + tan^2 theta
Now, let's solve for tan^2 theta:
tan^2 theta = 169/144 - 1
tan^2 theta = (169 - 144)/144
tan^2 theta = 25/144
To find the values of sin theta and cot theta, we can use the following trigonometric relations:
sin theta = 1 / √(1 + cot^2 theta) and cot theta = 1 / tan theta
Let's substitute the value of tan^2 theta into these equations:
sin theta = 1 / √(1 + (1 / tan^2 theta))
cot theta = 1 / tan theta
Substituting the value of tan^2 theta, we get:
sin theta = 1 / √(1 + (1 / (25/144)))
cot theta = 1 / tan theta
Simplifying further, we get:
sin theta = 1 / √(1 + (144/25))
cot theta = 1 / tan theta
sin theta = 1 / √(169/25)
cot theta = 1 / tan theta
sin theta = 1 / (13/5)
cot theta = 1 / tan theta
sin theta = 5/13
cot theta = 1 / tan theta
Therefore, the value of sin theta is 5/13, and the value of cot theta is 13/5.