Evaluate lim->4 sin(2y)/tan(5y)
Here is what I have so far. I am not sure the next steps. Can someone help me?
1. sin(2y)/(sin(5y)*cos(5y))
2. (sin(2y)*cos(5y))/sin(5y)
To evaluate the limit lim->4 sin(2y)/tan(5y), we can start by simplifying the expression.
1. sin(2y)/(sin(5y)*cos(5y))
We can rewrite the expression using trigonometric identities. The identity sin(2θ) = 2sin(θ)cos(θ) allows us to simplify the numerator:
2. [2sin(y)cos(y)] / (sin(5y)*cos(5y))
Now, let's simplify the denominator using the identity tan(θ) = sin(θ)/cos(θ):
3. [2sin(y)cos(y)] / [(sin(5y)/cos(5y))*cos(5y)]
The cos(5y) terms cancel out:
4. 2sin(y)cos(y) / sin(5y)
Now, we can factor out sin(y) from the numerator:
5. sin(y)(2cos(y)) / sin(5y)
We notice that there is a common term sin(y) in both the numerator and the denominator. We can simplify the expression further by cancelling out this common term:
6. 2cos(y) / sin(5y)
Now, let's evaluate the limit as y approaches 4:
lim->4 2cos(y) / sin(5y)
To evaluate this limit, we need to substitute the value of y into the expression. Since we are approaching the limit as y approaches 4, we substitute 4 into the expression:
2cos(4) / sin(5*4)
Using a calculator, we can find the values of cos(4) and sin(20) to evaluate the expression further.