Can someone help me with this prolem please?
Use the information given about the angle Ɵ to find the exact value of:
sin 2Ɵ.
1) sinƟ= 3/5, 0 < Ɵ < (π/2)
form a right-angled triangle in the first quadrant with the angle at the origin equal to Ɵ the opposite equal to 3 and the hypotenuse equal to 5
You should recognize the 3,4, 5 right-angled triangle
so cosƟ = 4/5
sin2Ɵ = 2sinƟcosƟ
= 2(3/5)(4/5)
= 24/25
check
if sinƟ = 3/5 then Ɵ = 36.8699
2Ɵ = 73.7398
sin 73.7398 = .96
and 24/25 = .96
Sure, I'd be happy to help you with this problem!
To find the exact value of sin 2Ɵ, we need to use the double-angle formula for sine. The double-angle formula states that sin 2Ɵ = 2sin Ɵ cos Ɵ.
From the given information, we know that sin Ɵ = 3/5, and 0 < Ɵ < (π/2). To find the value of cos Ɵ, we can use the Pythagorean identity, which states that sin^2 Ɵ + cos^2 Ɵ = 1.
Since sin Ɵ = 3/5, we can find the value of cos Ɵ as follows:
cos^2 Ɵ = 1 - sin^2 Ɵ
cos^2 Ɵ = 1 - (3/5)^2
cos^2 Ɵ = 1 - 9/25
cos^2 Ɵ = (25 - 9)/25
cos^2 Ɵ = 16/25
Taking the square root of both sides, we get:
cos Ɵ = ± √(16/25)
cos Ɵ = ± (4/5)
Since 0 < Ɵ < (π/2), we can conclude that cos Ɵ = 4/5, as the negative value of cos Ɵ would make Ɵ outside the given range.
Now we can substitute the values of sin Ɵ and cos Ɵ into the double-angle formula:
sin 2Ɵ = 2sin Ɵ cos Ɵ = 2 * (3/5) * (4/5) = 24/25
Therefore, the exact value of sin 2Ɵ is 24/25.
I hope this helps! Let me know if you have any further questions.