Find sin θ if cot θ = -2 and cos θ < 0.

Solution:

Given, cot θ = -2 and cos θ < 0.

We have to find sin θ.

Since, cos θ < 0 and cot θ = - 2, sine quantity must be positive and the quadrant will be the 2nd quadrant.

Using trigonometric identity,

1 + cot² (θ) = cosec² (θ)

Now, 1 + (-2)² = cosec² (θ)

5 = cosec² (θ)

We know, cosec (θ) = 1/sin (θ)

So, cosec² (θ) = 1 / sin² (θ)

Now, 5 = 1 / sin² (θ)

On cross multiplication,

5 × sin² (θ) = 1

sin² (θ) = 1/5

sin (θ) = ±1/√5

Since, theta lies in the second quadrant sine quantity will be positive.

sin (θ) =1/√5

Therefore, sin (θ) =1/√5

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Find sin θ if cot θ = -2 and cos θ < 0.

Summary:

If cot θ = -2 and cos θ < 0, sin (θ) = 1/√5.