How do you work this problem? Find the exact value of sin 2u, cos 2u, and tan 2u given that cot u=3 and 180<u<270
To find the exact values of sin 2u, cos 2u, and tan 2u, we first need to determine the values of sin u, cos u, and tan u. Given that cot u = 3 and 180 < u < 270, we can use the relationship between cotangent and tangent to find the value of tangent.
Cotangent is the reciprocal of tangent:
cot u = 1/tan u
So, in this case, 1/tan u = 3.
To solve for tangent, we can take the reciprocal of both sides:
tan u = 1/3
Now that we have the value of tangent (tan u), we can use that to find the values of sin u and cos u.
Tangent is defined as:
tan u = sin u / cos u
So, we can rewrite our equation as:
1/3 = sin u / cos u
We can then multiply both sides by cos u to isolate sin u:
sin u = (1/3) cos u
Since we know that 180 < u < 270, we know that sine is negative in the third quadrant. Therefore, sin u = -1/3.
Now, to find cos u, we can use the Pythagorean identity:
cos^2 u + sin^2 u = 1
Plugging in the values we know:
cos^2 u + (-1/3)^2 = 1
cos^2 u + 1/9 = 1
cos^2 u = 1 - 1/9
cos^2 u = 8/9
Taking the square root of both sides and considering that cos is negative in the third quadrant:
cos u = -√(8/9) = -2√2/3
Now that we have the values of sin u, cos u, and tan u, we can find sin 2u, cos 2u, and tan 2u.
The double-angle formulas for sine, cosine, and tangent are:
sin 2u = 2sin u cos u
cos 2u = cos^2 u - sin^2 u
tan 2u = (2tan u) / (1 - tan^2 u)
Plugging in the values we found:
sin 2u = 2(-1/3)(-2√2/3) = 4√2/9
cos 2u = (-2√2/3)^2 - (-1/3)^2 = 8/9 - 1/9 = 7/9
tan 2u = (2(1/3)) / (1 - (1/3)^2) = 2/3 / (1 - 1/9) = 6/7
So, the exact values are:
sin 2u = 4√2/9
cos 2u = 7/9
tan 2u = 6/7