Question about horizontal asymptotes and a trigonometric equation (unrelated!). Thanks so much!?

Question

Okay, hi! First off, how do you find a horizontal asymptote? I know that for a vertical asymptote, you just find the value that makes the denominator equal zero, right? Well what about the horizontal asymptote?

And I also have a question about the reduction formula with trigonometric equations.... I'm using the formula y = Asinx + Bcosx = √(a²+b²)sin(x + C), and everything comes out okay until I get to problems like these:

y = -3sinx - 5cosX

I got √(34)sin(x+2.1), but my book says that the answer is √(34)sin(x+4.2).

Similarly, for y = -√2 sinx - √7 cosx, I got 3sin (x + 2.1) and the book says 3sin (x + 4.2).

What am I doing wrong?

Thank you so much! I'll pick a best answer as soon as it lets me (I think 4 hours after I post the question)! Okay thanks!

Answer 1

For horizontal asymptotes, you need to determine whether the function has a limit as x->±∞. Suppose lim (x->-∞) f(x) = 3, then f(x)s a horizontal asymptote at y=3. Similarly, if lim(x->∞) f(x) = c (for some constant c) then f(x) has a horizontal asymptote at y=c. There may be an asymptote on one side, both, or neither; if there is an asymptote on both sides, they may have the same or different values.

sin (A+B) = sin A cos B + cos A sin B
So sin(x + C) = (cos C) sin x + (sin C) cos x.
y = -3 sin x - 5 cos x
= √(34) [(-3/√34) sin x + (-5/√34) cos x]
So we need C such that cos C = -3/√34 and sin C = -5/√34.
If you evaluate arccos(-3/√34) on your calculator you will get 2.11 to 2 d.p. However, this is between π/2 and π, so it is in the second quadrant and will have a positive value of sin C. We need the equivalent angle in the third quadrant, which will be 2π - 2.11 = 4.17 (2d.p.)

Similarly, for y = -√2 sin x - √7 cos x, we need cos C = -√2 / 3 and sin C = -√7 / 3. Since sin C and cos C are both negative we need an answer in the third quadrant. If you use your calculator to evaluate arcsin (-√7 / 3) you will get -1.08 (2d.p.), which is in the fourth quadrant; the corresponding angle in the third quadrant is π + 1.08 = 4.22 (2d.p.)

In summary, you're accepting the value your calculator gives you without considering whether you need to transform it into a different quadrant.