Improve Your Geometry Skills with Practice Problems

Evaluate ${\csc }^{-<!--\; -\; -->1}<!--\; \; -->(\sec <!--\; \; -->5)$ without using a calculator.

Just come across a question regarding sequential maximization and simultaneous maximization, and I do not recall whether there are any established conditions for the equivalence. Anyone has some idea?
$\underset{x}{max}\underset{y}{max}f(x,y)=\underset{y}{max}\underset{x}{max}f(x,y)?$

How to solve goniometric equation where function arguments differ?
${\cos }^2<!--\; \; -->x+\cos <!--\; \; -->2x+1=0$

My teacher said that the central angles of a circle are equal to the measure of the arc, but I don't understand on how this could possibly work.
Can someone please explain how this is possible?

My initial investment is $\mathrm{\$<!--\; \$\; -->}100$, and I earn $1\mathrm{\%<!--\; \%\; -->}$ interest per day. I can opt for any number of compoundings per day (if twice per day, then the interest rate per compounding period is $0.5\mathrm{\%<!--\; \%\; -->},$ and so on), but I have to pay $\mathrm{\$<!--\; \$\; -->}0.01$ each time my interest is compounded. After 365 days I will close the account.
What would this equation look like, and how should I include this to maximize my total deposit? How to generalize and figure out a good or optimal maximization?

Are partial derivatives of parametric surfaces always orthogonal?
For a surface $\mathbf{r}=\mathbf{r}(s,t)$ are the partial derivates ${\mathbf{r}}_s$ and ${\mathbf{r}}_t$ in general orthogonal?

$\sin <!--\; \; -->x+\sin <!--\; \; -->2x+\sin <!--\; \; -->3x=1+\cos <!--\; \; -->x+\cos <!--\; \; -->2x$ - solving a trigonometric equation

my intuition says I'm right, but I couldn't find or get a proof for it:

Suppose I have a vector function $\stackrel{\to <!--\; \to \; -->}{x}(t)\in <!--\; \in \; -->{\mathbb{R}}^n$. I am interested in finding ${\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}_t||\stackrel{\to <!--\; \to \; -->}{x}(t)|{|}_2$. For some reasons, this optimization problem is hard to solve, what I can solve easily, however, is ${\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}_t||\stackrel{\to <!--\; \to \; -->}{x}(t)|{|}_1$.

I know that 1-norm and 2-norm are equivalent in ${\mathbb{R}}^n$. Does this also include that the solution of these two problems are equivalent?

From origin walk halfway to (8, 6), turn 90 degrees left, then walk twice as far.

So, I have an arc that's part of a circle that I must "travel across".

The circle has a radius of 15 - a circumference of 94.248 (rounding). The arc length in question is equal to 15.708.

Now, normally, you could sort of walk along the arc, turning with it. But, if I could only travel straight, I need to know how I would get from one part of the arc to 15.708 units later in the arc of the circle (only being able to travel straight, or make 90 degree turns).

Solve $\sec <!--\; \; -->x+\tan <!--\; \; -->x=4$
Find x for $0<x<2\mathrm{\pi <!--\; \pi \; -->}$
Eventually I get
$\cos <!--\; \; -->x=\frac{8}{17}$
$x={61.9}^{\circ <!--\; \circ \; -->}$
The answer I obtained is the only answer, another respective value of x in 4-th quadrant does not solve the equation, how does this happen? I have been facing the same problem every time I solved this kind of trigonometric equation.

Here are the five postulates:
1. Each pair of points can be joined by one and only one straight line segment.
2. Any straight line segment can be indefinitely extended in either direction.
3. There is exactly one circle of any given radius with any given center.
4. All right angles are congruent to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than two right angles.

Questions:
1. These to me sounds more like something that shouldn't require proving... does it?
2. Why is it important to stress things that are obvious? For example, what other answers can you get when extending a line segment other than it can be extended indefinitely in either direction?
3. Similarly, what space can allow two circle of the same radius and center to be not the same?
4. Saying all right angles are congruent ... isn't that the same as saying all 64.506 degree angles are congruent? Isn't it ANY angle are congruent if they are the same degrees measures from the same reference point (say x-axis)?
5. Why do we need the 5th postulate?

Solve equation for t
$s=2\ln <!--\; \; -->|\tan <!--\; \; -->(t)+\sec <!--\; \; -->(t)|$

I am writing a computer program where I have $x$ real positive varying in the domain $[\sqrt{U},U]$. I want the value of $x$ which maximizes:
$(1+\sqrt{U})-<!--\; -\; -->\frac{\sqrt{U}-<!--\; -\; -->1}{U-<!--\; -\; -->\sqrt{U}}x-<!--\; -\; -->\frac{U}{x}$
("visually", it seems to be a bit above $2\sqrt{U}$) what is the best way to find it exactly?

What is the height of a circular cone with surface line ("Mantellinie") s, which has the maximal volumina? My problem here is, that I am not really sure what surface line is..is this the same thing as just to say surface?

Prove that angle bisectors of a triangle are concurrent using vectors. Also, find the position vector of the point of concurrency in terms of position vectors of the vertices.

I solved this without using vectors to get some idea. I am not sure how to prove it using vectors. I don't want to use vector equations for straight lines and then find the point of concurrency. That's like solving using coordinate geometry.

Let $\stackrel{\to <!--\; \to \; -->}{a},\stackrel{\to <!--\; \to \; -->}{b},\stackrel{\to <!--\; \to \; -->}{c}$ represent the sides $A,B,C$ respectively.

The angle bisectors are along ${\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{a}}{|\stackrel{\to <!--\; \to \; -->}{a}|}}+{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{b}}{|\stackrel{\to <!--\; \to \; -->}{b}|}},{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{b}}{|\stackrel{\to <!--\; \to \; -->}{b}|}}+{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{c}}{|\stackrel{\to <!--\; \to \; -->}{c}|}},{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{c}}{|\stackrel{\to <!--\; \to \; -->}{c}|}}+{\displaystyle \frac{\stackrel{\to <!--\; \to \; -->}{a}}{|\stackrel{\to <!--\; \to \; -->}{a}|}}$

Let the sides $AB,BC,CA$ be $x,y,z$. Let $AD$ be one of the angular bisector.
$\frac{BD}{CD}=\frac{x}{z}$
Hence
$D=\frac{x\stackrel{\to <!--\; \to \; -->}{c}+z\stackrel{\to <!--\; \to \; -->}{b}}{x+z}$
What should be the next step? Or is there a better method?

For reference: In the drawing, T is the point of tangency, $LN||AT$, $OH=4$ and $L{N}^2+A{M}^2=164$ . Calculate HN.

Show that,
$2\arctan <!--\; \; -->\left(\frac{1}{3}\right)+2\arcsin <!--\; \; -->\left(\frac{1}{5\sqrt{2}}\right)-<!--\; -\; -->\arctan <!--\; \; -->\left(\frac{1}{7}\right)=\frac{\mathrm{\pi <!--\; \pi \; -->}}{4}$
There is a mixed of sin and tan, how can I simplify this to $\frac{\mathrm{\pi <!--\; \pi \; -->}}{4}$
We know the identity of $\arctan <!--\; \; -->\left(\frac{1}{a}\right)+\arctan <!--\; \; -->\left(\frac{1}{b}\right)=\arctan <!--\; \; -->\left(\frac{a+b}{ab-<!--\; -\; -->1}\right)$

Pythagorean Theorem intuition
I actually have a degree in Pure Mathematics, and this has always bugged me. Like a tiny, hardly-noticeable stone in your shoe you just can't get out.
Despite being at the core of geometry, arithmetic, and infinity, despite being able to prove it a dozen different ways, and despite it being the first "formula" anyone learns and one of the oldest known mathematical ideas - I can only see shadows and silhouettes of its truth.
Looked at from the point of view of geometry, I can see the statement plainly as a statement regarding the intrinsic notions of angle and area (using the proof of similar right-angle triangles) - however, the last step requires a correspondence between the notions of length and area to make complete (namely, the area of a shape increases quadratically as its scale increases linearly). This has always bothered me, since, the Theorem is an idea solely between angle and length.
There are, of course, simple proofs that don't depend on the notion of area (ex: Pythagorean Theorem Proof Without Words 6), however, sadly, they don't further my intuition, despite being clever/cute.
Staring at the statement blank in the face, I understand it far less intuitively then I do half of the ideas in algebra, analysis, number theory, etc.

Is there a non-right triangle which corresponding circumcenter is lying on one of its sides?