Answer:
Explanation:
We know that:
cosec A - cot A = 1/3
We can write cosec A as 1/sin A and cot A as cos A/sin A, so we can substitute and simplify:
1/sin A - cos A/sin A = 1/3
Multiplying both sides by 3sin A, we get:
3 - 3cos A = sin A
We can square both sides to eliminate the sin A term, since sin^2 A + cos^2 A = 1:
(3 - 3cos A)^2 = sin^2 A = 1 - cos^2 A
Expanding and simplifying, we get:
9 - 18cos A + 9cos^2 A = 1 - cos^2 A
10cos^2 A - 18cos A + 8 = 0
Dividing both sides by 2, we get:
5cos^2 A - 9cos A + 4 = 0
This is a quadratic equation, which we can solve using the quadratic formula:
cos A = [9 ± sqrt(9^2 - 4(5)(4))]/(2(5))
cos A = [9 ± sqrt(41)]/10
Since cosec A is the reciprocal of sin A, we can find its value using the Pythagorean identity:
sin^2 A + cos^2 A = 1
sin^2 A = 1 - cos^2 A
sin A = ± sqrt(1 - cos^2 A)
Since cosec A is positive, we take the positive square root:
cosec A = 1/sin A = 1/[+/- sqrt(1 - cos^2 A)]
We can substitute the value of cos A that we found earlier:
cosec A = 1/[+/- sqrt(1 - (9 ± sqrt(41))^2/100)]
Evaluating the expression with the plus and minus signs, we get two possible values:
cosec A ≈ 1.7641 or cosec A ≈ -2.5245
Since cosec A is positive, we choose the first value:
cosec A ≈ 1.7641
Therefore, cosec A ≈ 1.7641.