Let fx9sinx4sinx6cosx then fx 544sinx6cosx2 the equation of...

Let f(x) = (9sinx)/(4sinx + 6cosx). Then f'(x) = 54/(4sin(x) + 6cos(x))^2. The equation of the tangent line to y = f(x) at a = π/3 can be written in the form y = mx + b, where m = and b = .

Transcript

00:01 The way i'm reading this problem is you need to know pi over 3 and where it's at on the unit circle and the ordered pair.

00:09 So the ordered pair of pi over 3 is 1⁄2.

00:14 So then the next thing to acknowledge is what is, well, how about this? let's make sure that we know that sine went to the wrong one.

00:25 So cosine of pi over 3 equals 1 half.

00:30 And sine of pi over 3 is the y coordinate is root 3 over 2.

00:36 So when they ask you to do the tangent line, first thing you want to do is figure out what f of pi over 3 is, and to speed this along, so i don't have to type out so much, is 9 sine of pi over 3 would be 9 times root 3 over 2, over 4 sine of pi over 3 plus 6 cosine, which would be 1⁄2.

01:02 So just doing a quick, maybe what i would do is multiply the top and bottom by two.

01:08 So that'll get rid of the fractions inside of fractions.

01:12 So like these twos would cancel.

01:14 These twos would cancel.

01:15 So what i'm left with is nine root three on top over four root three plus six.

01:21 Now i'm going to leave it like this because they don't ask you to rationalize the denominator.

01:28 It's kind of gross if you're looking at it this way.

01:32 So this is my y coordinate when x equals pi over three.

01:37 This is my y coordinate of that.

01:41 So i circled it and read it because it's not my final answer.

01:44 So now i need to do f prime of pi over three, which as i'm looking at this problem, we have 54 on top over four times sine is root 3 over 2 plus 6 times 1 half.

02:04 But that needs to be squared.

02:07 So i don't know if you're allowed to just leave your answer like this, but it's kind of gross if you do.

02:13 You could simplify a little bit, but it's not going to be that helpful.

02:17 Like i'm thinking about rewriting us 54 over 2 root 3 plus 3 squared, but even then you could simplify a little bit because what does it mean to be squared is multiplied by itself.

02:34 Let me erase this and change it to times 2 root 3 plus 3.

02:40 And if you're trying to simplify that, there's a lot of work...

Question

Let f(x) = (9sinx)/(4sinx + 6cosx). Then f'(x) = 54/(4sin(x) + 6cos(x))^2. The equation of the tangent line to y = f(x) at a = π/3 can be written in the form y = mx + b, where m = and b = .

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