Can someone help me with this problem please? Thank-you to those who helped me with the previous ones and I understand those. But, I tried this one and I don't understand it.
Use the information given about the angle Ɵ to find the exact value of:
sin 2Ɵ.
1) secƟ= 3, SinƟ > 0
Isnt the sin2theta=sintheta*cosTheta?
You are given cosTheta=1/3, so use
sin^2 theta + cos^2 theta=1 to find sinTheta.
ok thank-you. i figured it out
Of course! I'll be happy to help you with this problem.
In order to find the exact value of sin 2Ɵ, we'll need to use trigonometric identities and the given information.
First, let's recall the double-angle identity for sine:
sin 2Ɵ = 2sin Ɵ cos Ɵ
Based on the given information, we know that secƟ = 3 and SinƟ > 0.
Since secƟ is the reciprocal of cosƟ, we can find cosƟ by taking the reciprocal of 3:
cosƟ = 1/3
Now, to find sin Ɵ, we'll use the Pythagorean identity:
sin^2 Ɵ + cos^2 Ɵ = 1
Plugging in the value of cosƟ, we get:
sin^2 Ɵ + (1/3)^2 = 1
sin^2 Ɵ + 1/9 = 1
sin^2 Ɵ = 1 - 1/9
sin^2 Ɵ = 8/9
Taking the square root of both sides, we find:
sin Ɵ = √(8/9)
Since sinƟ > 0, we know that Ɵ is in either the first or second quadrant.
Now, let's substitute these values back into the double-angle identity:
sin 2Ɵ = 2sin Ɵ cos Ɵ
sin 2Ɵ = 2(√(8/9))(1/3)
= 2√(8/27)
= 2(√8/√27)
= 2(2√2/3√3)
= (4√2)/(3√3)
So, the exact value of sin 2Ɵ is (4√2)/(3√3).
I hope this explanation helps you understand how to solve the problem. Let me know if you have any further questions!