Question

(06.03 HC)

Let sin8= and <0<.
2
2√2
5
Part A: Determine the exact value of cos 20. (5 points)
0
Part B: Determine the exact value of sin
2
(5 points)

Answer 1

Part A:

To determine the exact value of cos 20, we can use the trigonometric identity cos^2θ + sin^2θ = 1. Since sin^2θ = (sin 8)^2 = (2√2/5)^2 = 8/25, we can solve for cos^2θ as follows:

cos^2θ = 1 - sin^2θ

cos^2θ = 1 - 8/25

cos^2θ = 17/25

Taking the square root of both sides, we get:

cosθ = ±√(17/25)

Since 0 < θ < π/2 (given that θ is acute), cosθ is positive. Therefore:

cosθ = √(17/25)

cosθ = √17/5

So, the exact value of cos 20 is √17/5.

Part B:

To determine the exact value of sin (2θ), we can use the double-angle formula for sine, which states that sin (2θ) = 2sinθcosθ. Given that sin 8 = 2√2/5 and cosθ = √17/5 (from Part A), we can substitute these values into the formula:

sin (2θ) = 2(sin 8)(cosθ)

sin (2θ) = 2(2√2/5)(√17/5)

sin (2θ) = (4√2√17)/25

sin (2θ) = (4√34)/25

So, the exact value of sin (2θ) is (4√34)/25.

Explanation: