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, [tex]\frac{3\pi}{4}[/tex], 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+[tex]\sqrt{3}i)^{4}[/tex]

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, [tex]\frac{3\pi}{4}[/tex], 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+[tex]\sqrt{3}i)^{4}[/tex]

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!

 

[Architeuthis Dux]

Hello,

I would suggest breaking down each concept into smaller parts and tackling them one by one. For example:

6.c. To verify the identity sinx(cot x + tan x) = sec x, you can start by using the definition of cotangent and tangent (cot x = cos x / sin x and tan x = sin x / cos x). Then, substitute these values into the left side of the equation and simplify until you get the right side of the equation. This will help you understand the steps and concepts involved in verifying trigonometric identities.

17. To find two other polar coordinate representations of the point [-2, \frac{3\pi}{4}], you can start by converting the point to rectangular coordinates using the formulas x = r cos θ and y = r sin θ. Then, you can try different values of r (positive and negative) to get different polar coordinate representations of the point.

22.b. DeMoivre's theorem states that (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). So, for the given problem, you can first find the polar form of the complex number (-3 + √3i), and then use DeMoivre's theorem to find the indicated power.

24.a. To find parametric equations for a line with slope 2 and passing through the point (-10,8), you can use the formula y = mx + b, where m is the slope and b is the y-intercept. In this case, m = 2, so you can use the point-slope form of a line (y - y1 = m(x - x1)) to find the equation of the line. Then, you can convert this equation into parametric form by letting x = t and solving for y in terms of t.

30. For this problem, you can use vector addition to find the true velocity of the motorboat. The velocity of the river is given as 10 mph in the east direction, so you can represent this as a vector (10,0). The velocity of the motorboat relative to the water is given as 20 mph in a direction 60 degrees north from the shore, so you can represent this as a vector (10√3,10). Then, to find the true velocity of the motorboat, you can add these two vectors together. The magnitude of the resulting vector