Hon. Trig & Precalc Exam Review - Concepts

[jacksonpeeble]

I've been studying for our final exam throughout this weekend, and I realized that there is a lot that I don't remember. So, instead of asking each individual question, I'm going to try to ask one from each concept that I don't understand. Quite a few of them I have absolutely no idea how to start, which is why I might not show work (sorry), but I did already complete a portion of the review (if that's any consolation ). Anyhow, any help would be greatly appreciated, as always!

Homework Statement

6.c. Verify the following identities: sinx(cot x + tan x) = sec x

17. Give two other polar coordinate representations of the point [-2, \frac{3\pi}{4}, one with r<0 and the other with r>0.

22.b. Find the indicated power using DeMoivre's Theorem. Have your answer in polar form. (-3+\sqrt{3}i)^{4}

24.a. Find parametric equations for the line with the give properties. Slope=2, passing through (-10,8)

30. A straight river flows east at a speed of 10 mph. A boater starts at the south shore of the river and heads in a direction 60 degrees north from the shore. The motorboat has a speed of 20 mph relative to the water.
a. Express the velocity of the river as a vector in component form.
b. Express the velocity of the motorboat relative to the water as a vector in component form.
c. Find the true velocity of the motorboat as a vector.
d. Find the true speed and direction of the motorboat.

46. The sum of the first three terms of a geometric series is 52, and the common ratio is 3. Find the first term.

The Attempt at a Solution

6.c. I probably use tan=sin/cos and sec=1/cos in there somewhere...

17. Hmm...

22. DeMoivre's theorem squared the first term and multiplies the second. However, this obviously won't work directly on the coordinates given, so I need the preliminary operation.

24. Well, I know the slope, but how do I set up the equations?

30. I drew a pretty little diagram, and that's about it.

46. Does this use sigma?Thanks again! If you can only contribute on one, that's fine, too; all help is appreciated!

 

[Mark44]

I've been studying for our final exam throughout this weekend, and I realized that there is a lot that I don't remember. So, instead of asking each individual question, I'm going to try to ask one from each concept that I don't understand. Quite a few of them I have absolutely no idea how to start, which is why I might not show work (sorry), but I did already complete a portion of the review (if that's any consolation ). Anyhow, any help would be greatly appreciated, as always!

Homework Statement

6.c. Verify the following identities: sinx(cot x + tan x) = sec x

17. Give two other polar coordinate representations of the point [-2, \frac{3\pi}{4}, one with r<0 and the other with r>0.

22.b. Find the indicated power using DeMoivre's Theorem. Have your answer in polar form. (-3+\sqrt{3}i)^{4}

24.a. Find parametric equations for the line with the give properties. Slope=2, passing through (-10,8)

30. A straight river flows east at a speed of 10 mph. A boater starts at the south shore of the river and heads in a direction 60 degrees north from the shore. The motorboat has a speed of 20 mph relative to the water.
a. Express the velocity of the river as a vector in component form.
b. Express the velocity of the motorboat relative to the water as a vector in component form.
c. Find the true velocity of the motorboat as a vector.
d. Find the true speed and direction of the motorboat.

46. The sum of the first three terms of a geometric series is 52, and the common ratio is 3. Find the first term.

The Attempt at a Solution

6.c. I probably use tan=sin/cos and sec=1/cos in there somewhere...

Yes, absolutely. I would make these replacements and see if I could make the left side look like the right side.

17. Hmm...

I think all you need to do is find a different angle representation that gets you to the same point on the unit circle. For example, (1, pi/4) can be represented as (-1, 5pi/4) or as (1, 9pi/4).

22. DeMoivre's theorem squared the first term and multiplies the second. However, this obviously won't work directly on the coordinates given, so I need the preliminary operation.

Rewrite the vector in polar form, and then you can use DeMoivre's theorem on it.

24. Well, I know the slope, but how do I set up the equations?

Get the equation of the line first in y and x. The simplest way would be with this formula: y - y₀ = m(x - x₀). After that, you can parametrize using x = t, y = mt + b. If that's not what the problem is looking for, you can parametrize it using the vector sum of the vector to the given point + t times the slope.

30. I drew a pretty little diagram, and that's about it.

This one will take a little work. Take a look at the diagram you drew and see if you can make a start at equations that represents the quantities involved.

46. Does this use sigma?

You don't need it, since you're dealing with just the first three terms. And the series is a geometric series, so each term is (in this case) 3 times the previous term.

Thanks again! If you can only contribute on one, that's fine, too; all help is appreciated!