Question

If cosec θ + cot θ = $\frac{1}{3}$, find cos θ and determine the quadrant in which θ lies.

Answer 1

cosec θ + cot θ = 1/3 .........(1)

we know, cosec²θ - cot²θ = 1

or, (cosecθ - cotθ)(cosecθ + cotθ) = 1

from equation (1),

or, (cosecθ - cotθ) × 1/3 = 1

or, cosecθ - cotθ = 3 ........(2)

solve equations (1) and (2),

2cosecθ = 1/3 + 3

cosecθ = 5/3 => sinθ = 3/5 = p/h

so, p = 3 and h = 5 then, b = √(5² - 3²) = ±4

now, cosθ = b/h = ±4/5

hence, cosθ = ±4/5

θ lies in 1st or 3rd quadrant.

Answer 2

Answer:

Step-by-step explanation:

$cosec\theta+cot\theta=\frac{1}{3}$

$\frac{1}{sin\theta}+\frac{cos\theta}{sin\theta}=\frac{1}{3}$

$\frac{1+cos\theta}{sin\theta}=\frac{1}{3}$

$\frac{2{cos}^2\frac{\theta}{2}}{2sin\frac{\theta}{2}.cos\frac{\theta}{2}}=\frac{1}{3}$

$cot\frac{\theta}{2}=\frac{1}{3}$

$tan\frac{\theta}{2}=3$

$cos\theta=\frac{1- {3}^2}{ 1+ {3}^2}$

$cos\theta=\frac{-8}{10}\\\\cos\theta=\frac{-4}{5}\\\\\theta\:lies\:either\:second\:or\:third\:quadrant.$