Question

If cos ϴ = (-3)/( 5) and π < ϴ < 3π/2 , then find the values of remaining trigonometric functions and hence evaluate(cosecϴ + cotϴ)/(secϴ - tanϴ)

Answer 1

AnsWer :

1 / 6.

SolutioN :

$\tt \longmapsto cos \theta = \dfrac{ - 3}{5}$

We know,

• Sin² A + Cos² A = 1.

$\tt \longmapsto {sin}^{2} \theta + {cos}^{2} \theta = 1.$

Putting the value of Cos ϴ.

$\tt \longmapsto {sin}^{2} \theta + {( \frac{ - 3}{5}) }^{2} = 1.$

$\tt \longmapsto {sin}^{2} \theta = 1 - {( \frac{ - 3}{5}) }^{2}$

$\tt \longmapsto {sin}^{2} \theta = 1 - \dfrac{ 9}{25}$

$\tt \longmapsto {sin}^{2} \theta = \dfrac{25 - 9}{25}$

$\tt \longmapsto {sin}^{2} \theta = \dfrac{16}{25}$

$\tt \longmapsto sin \theta = \pm \dfrac{4}{5}$

$\rule{120}3$

Now, We have value of sin ϴ , cos ϴ

★ Let's Find tan ϴ.

• We can write as tan ϴ = sin ϴ / cos ϴ.

$\tt \longmapsto tan\theta = \dfrac{sin \theta}{cos \theta}$

$\tt \longmapsto tan\theta = \dfrac{ \frac{4}{5} }{ \frac{ 3}{5} }$

$\tt \longmapsto tan\theta = \pm\dfrac{4}{3}$

°•° Note : cos ϴ lies on π < ϴ < 3π/2 .

• Quadrant third.
• only ( tan ϴ and cot ϴ are positive )
• More information in attachment ↑

So,

• Sin ϴ = - 4 / 5.
• cos ϴ = - 3 / 5.
• tan ϴ = 4 / 3.
• cosec ϴ = - 5 / 4.
• sec ϴ = - 5 / 3.
• cot ϴ = 3 / 4.

$\rule{ 90}2$

A/Q,

$\tt \longmapsto \dfrac{(cosec \theta + cot \theta)}{(sec \theta - tan\theta)}$

→ - 5 / 4 + 3 / 4 // - 5 / 3 - 4 / 3

→ - 1 / 2 * - 1 / 3.

→ 1 / 6.

Therefore, the required value is 1 / 6.