Question

$cos\theta =\sqrt2/3, 3\pi /2\leq \theta \leq 2\pi , find csc\theta, find cot\theta$

Answer 1

Step-by-step explanation:

Given:-

Cos θ = √2/3 , 3π/2 ≤ θ ≤ 2π

To find:-

Find the values of Cosec θ and Cot θ ?

Solution:-

Given that Cos θ = √2/3

On squaring both sides

(Cos θ )^2 = (√2/3)^2

=>Cos^2 θ = 2/9

On subtracting from 1 both sides then

=>1 - Cos^2 θ = 1-(2/9)

=>1 - Cos^2 θ = (9-2)/9

=>1 - Cos^2 θ = 7/9

We know that

Sin^2 A + Cos^2 A = 1

=>Sin^2 θ = 7/9

=>Sin θ = √(7/9)

=>Sin θ = √7/3

We know that

1/ Sin θ = Cosec θ

Cosec θ = 1/(√7/3)

=>Cosec θ = 3/√7 ------(1)

(or)

=>Cosec θ = (3×√7)/(√7×√7)

=>Cosec θ = 3√7/7

On squaring (1) both sides

=>Cosec^2 θ = (3/√7)^2

=>Cosec^2 θ = 9/7

We know that

Cosec^2θ - Cot^2 θ = 1

=>Cot^2 θ = Cosec^2 θ - 1

=> Cot^2 θ = (9/7)-1

=>Cot^2 θ = (9-7)/7

=>Cot^2 θ = 2/7

=> Cot θ = √(2/7)

(or)

Cot θ = Cos θ/ Sin θ

=> Cot θ = (√2/3)/(√7/3)

=>Cot θ = (√2/3)×3/√7)

=>Cot θ = √2/√7

=>Cot θ =√(2/7)

Answer:-

Cosec θ =3/√7 or (3√7)/7

Cot θ = √(2/7)

Used formulae:-

• Sin^2 A + Cos^2 A = 1

• Cosec^2 A - Cot^2 A = 1

• Cosec A = 1/Sin A

• Cot A = Cos A / Sin A