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wolffyluna · 1 year ago

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[NB: I am partway through The Legacy.]

Not to solve a love triangle with polyamory, but Drizzt/Catti-Brie/Wulfgar has legs. The polyamory doesn't solve the problem easily, but it does solve it interestingly.

Because there's a thing where Catti-Brie is canonically making a choice between these two.And Wulfgar is intensely jealous of Drizzt, but it's part of the thing where ~Wulfgar is from a culture where women should be submissively barefoot in the kitchen, but the reason he likes Catti-Brie is because she is not like that, so he gets torn between what he wants and what he thinks he wants/ the scripts he has. Meanwhile Drizzt is from a misandrist society, and his own family actually contained unethical polyamory. (And like. He observes that most of the non-disastrous non-drow cultures are monogamous, so that's evidence in favour of it, but also the polyamory really wasn't the problem in his family, it was the ~power dynamics).

What I want is Drizzt having a private conversation with Wulfgar along the lines of: he wants them both to be happy. He would like to have a relationship with Catti-Brie. But if it ever becomes a competition, he is out of there. (And general stuff of not wanting to be Zaknafein.)

Also, separately, I want it to start as a V that ends up becoming a triad, but the way people realise this has happened is that Wulfgar ends up being jealous that Catti-Brie is getting Drizzt's attention, going '... isn't that backwards?' and generally being 'pythagoran theorem, volume of a cone' about his own emotional response

#legend of drizzt #ideas #wolffy reads drizzt

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enetarch-math · 7 years ago

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Chapter 7.1 & 2 - Trigonometry

Triangles are everywhere. They are the worlds strongest structure. They form boxes and circles; cubes and spheres. They are used to represent the location of 3 objects in relationship to each other. So, in this next segment of the Math blog, we will look at how algebraic equations use trigonometric functions.

Let’s start by reviewing the basics on trigonometry.

Triangles

A triangle always has 3 sides. Always!

The sum of the interior angles of a triangle always equal 180 degrees.

There are 3 special types of triangles:

Right Triangles .. A triangle with a 90 degree corner

Isosceles Triangles .. 2 angles and 2 sides are the same.

Equilateral Triangles .. 3 angles and 3 sides are the same.

Acute and Obtuse triangles are not special, as they can be turned into 2 right triangles. They describe how a triangle is viewed, and state that one of the angles in the triangle has an angle greater than 90 degrees.

An acute triangle’s longest edge is displayed as the base.

An obtuse triangle’s longest edge is not the base.

The length of any side of a right triangle can be found as long as you have 2 things

The lengths of 2 sides of a triangle

A length of a side and an angle

All 3 angles created by the legs of a triangle can be found with 2 pieces of information.

The lengths of 2 sides of a triangle

A length of a side and an angle

The angles from 2 corners

Terminology

As mentioned, a triangle has 3 sides (or 3 segments). These sides have specific names and letter abbreviations. They are most often used in right triangles to describe the angle and sides being worked with.

Hypotenuse .. (h) .. is the longest edge along a right triangle.

Opposite Side .. (o) .. is the edge opposite the angle in a right triangle.

When drawn on graph paper, this segment of a triangle runs along the (y-axis).

Adjacent Side .. (a) .. is the edge that runs adjacent to the angle in a right triangle.

When drawn on graph paper, this segment of a triangle runs along the (x-axis).

The right angle is represented by a box in the bottom right corner of the triangle.

The angle we will be considering through this series of posts, is in the bottom left corner, represented as a pie slice or wedge.

Equations

Before we get into the functions of triangles, let’s look at some basic equations that are used with right triangles.

The equation of a line:

y = mx +b

(m) represents the (rise / run) of line (or segment) drawn as the hypotenuse (h) of the triangle.

(x) represents that length or run of the adjacent (a) side of a right triangle.

(y) represents the height or rise of the opposite (o) side of a right triangle.

Pythagorean’s Theorem states that you can find the length of the hypotenuse (h) if you know the lengths of the adjacent (a) and opposite (o) sides of a triangle.

h^2 = a^2 + o^2

These equations describes a circle and an ellipse respectively:

r^2 = (x - i)^2 + (y - j)^2 1 = ((x - i)^2) / a^2 + ((y - j)^2) / b^2

Note: if you look closely at these equations and pythagorean’s theorem, you will see how the a^2 and o^2 have been extended. Example: a^2 = (x - i)^2

These equations are they hyperbolic functions of a circle and ellipse respectively:

r^2 = (x - i)^2 - (y - j)^2 1 = ((x - i)^2) / a^2 - ((y - j)^2) / b^2

Note: the only difference is the ( - ) minus sign in the equation.

Functions

There are 3 types of functions that work on triangles. And, each function returns a value between 0 and 1.

n = Sin (angle)

n = Cos (angle)

n = Tan (angle)

Note: a function takes 1 value and returns 1 value. It can’t return 2 values, otherwise, it’s not a function. Example: y = F(x)

Note: There are other functions, but they are versions of these 3 functions. They will be covered briefly, while most of the time is spent reviewing these 3 functions.

These 3 functions take in a value (angle) which is a number

between 0 and 359 degrees

between 0 and 2 PI

between 0 to 99 radians

Note: these numbers describe a relationship between two segments that construct a triangles adjacent (a) and opposite (o) sides. More on this in a minute.

When you see a right triangle, it is suggested that you draw it as shown. Eventually, your brain will comprehend its parts no matter how the triangle is rotated, reversed, or inverted. So, for now, we will discuss right triangles from this orientation.

Sin (angle) = o / h

Cos (angle) = a / h

Tan (angle) = o / a

It is recommended that you memorize these relationships.

Exercises

1) Create 4 right triangles

Take a piece of letter size paper (8.5 x 11) and create a square.

Fold 1 corner of the paper to the opposite of the paper to create a triangle.

Actually you have 2 triangles, one on top of the other.

Crease the paper

Fold the remaining piece of paper, that creates a rectangle vs a square, and tear it off.

When you unfold the 2 triangles to form a square.

Take the rectangle and crease it between 2 opposite corners

Seperate all the triangles

2) Mark the 90 degree angle on a triangle

3) Mark the hypotenuse, adjacent and opposite edges on 1 triangle

4) Write the measurements of the angles of 2 different triangles (one from the square, one from the rectangle) in the corner on their respective triangles.

Do they add up to 180 degrees?

Does one of the triangles have a 45 degree angle?

Which triangle is an Isosceles triangle?

Are any of the triangles obtuse or acute?

5) Write the measurements of the sides of 2 different triangles (one from the square, one from the rectangle), and perform the following algebraic expressions with their values.

h^2 = a^2 + o^2

does the sin (angle) = o / h

does the cos (angle) = a / h

does the tan (angle) = o / a

does the o = h * sin (angle)

does the a = h * cos (angle)

does the o = a * tan (angle)

6) Using graph paper, and a ruler, create 10 different triangles where the adjacent (a) and opposite (o) sides add up to 15 centimeters. Then use the equations above to figure out what the hypotenuse (h) and angle would be.

7) using graph paper and a ruler, create 10 different triangles where the hypotenuse (h) is 10 centimeters long. Then use the equations above to figure out what the adjacent (a), opposite (o) and angle would be.

#precalc #trigonometry #right triangles #pythagorans theorem #sine #cosine #tangent

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