Question

Sin 2α, cos 2α, если sinα=4/5: 0<α<π/2

Answer 1

sinα=4/5

cosα= √{1-sin²α} = √{1-16/25} = √9/25 = 3/5

sin 2α = 2 sin α · cos α = 2*4/5 * 3/5 = 24/25

cos 2α = cos² α - sin² α = 9/25 - 16/25 = -7/25

Answer 2

To find the values of sin 2α and cos 2α given that sin α = 4/5 and 0 < α < π/2, we can use the double-angle trigonometric identities:

1. sin 2α = 2sin α * cos α
2. cos 2α = cos² α - sin² α

First, we need to find the value of cos α using the given information that sin α = 4/5. Since 0 < α < π/2, we know that α lies in the first quadrant, and cos α is positive.

We can use the Pythagorean identity to find cos α:

cos² α + sin² α = 1

Plugging in sin α = 4/5:

cos² α + (4/5)² = 1 cos² α + 16/25 = 1 cos² α = 1 - 16/25 cos² α = 9/25

Taking the square root of both sides (since cos α is positive):

cos α = √(9/25) = 3/5

Now that we have the value of cos α, we can calculate sin 2α and cos 2α using the double-angle trigonometric identities:

1. sin 2α = 2sin α * cos α sin 2α = 2 * (4/5) * (3/5) sin 2α = 24/25

2. cos 2α = cos² α - sin² α cos 2α = (3/5)² - (4/5)² cos 2α = 9/25 - 16/25 cos 2α = -7/25

So, the values of sin 2α and cos 2α are:

sin 2α = 24/25 cos 2α = -7/25