Mathematics College

Given cosθ = 2/5 ​and angle θ is in Quadrant IV, what is the exact value of sinθ in simplest form? Simplify all radicals if needed.

Answers

Answer 1

The exact value of sinθ in simplest form, with cosθ = 2/5 and θ in Quadrant IV, is -√21/5.

To find the exact value of sinθ, we can use the Pythagorean identity: sin²θ + cos²θ = 1. Since we know the value of cosθ, we can substitute it into the equation and solve for sinθ.

Given that cosθ = 2/5 and θ is in Quadrant IV, we know that the cosine is positive in Quadrant IV, and the sine is negative. Let's proceed with the calculations:

sin²θ + cos²θ = 1

sin²θ + (2/5)² = 1 (Substituting cosθ = 2/5)

sin²θ + 4/25 = 1 (Simplifying)

sin²θ = 1 - 4/25 (Subtracting 4/25 from both sides)

sin²θ = 21/25 (Simplifying)

Taking the square root of both sides, considering that sinθ is negative in Quadrant IV:

sinθ = -√(21/25)

Since we want to express the answer in simplest form, we can simplify the radical:

sinθ = -√21/√25

The square root of 25 is 5, so we can simplify further:

sinθ = -√21/5

Therefore, the exact value of sinθ in simplest form, given that cosθ = 2/5 and θ is in Quadrant IV, is -√21/5.

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