Submitted by chrisws on

Math function TAN

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## Comments

MGA replied on Permalink

## Trig Lesson two

see COS for Trig lesson one

see ATAN for a maybe fun demo involving TAN and ATAN

Trigonometry lesson two.

We will try this from a txt file instead.

Review:

Trigonometry is the study of right triangles,

those that have one 90 degree angle AKA Pi/2 in radians.

These right triangles are infinite in a rectilinear coordinate system

with x,y referring to screen position in relation to an origin point (0,0).

In SmallBASIC the y axis increases from top to bottom

and in SmallBASIC the angle measures used for arguments to SIN

and COS and the other Trig Angles are expected to be in radian

measure as opposed to degree measure.

This is a standard conflict and source of confusion from what you might

learn in math class and doing stuff on a computer.

For all triangles that have all 3 angles the same,

their legs will have exactly the same ratios between themselves.

Feed a trig function an angle and it will return the special ratio

between the sides of the triangle that is universal to all triangles

that have the same equal angle measures.

Feed the COS function an angle and it will return

the universal ratio for the leg adjacent divided by the hypotenuse.

( Adjacent means next to or attached in the case of angles.)

Some more information gained from study or right angles:

The COS for one acute angle in a right triangle is

the SIN for the other acute angle.

For 45 degree angle (Pi/4 radians) the two legs are the same therefore

SIN(RAD(45 degrees)) = COS(RAD(45 degrees)) = .7071...

notice if that is squared =.5 exactly 1-COS(angle)^2=SIN(angle)^2

As the angle size approaches 0, COS(angle) > 1 and SIN(angle) >0

At angle = 0 you have more a line segment than a triangle.

Similarly as angle approaches 90 degrees = Pi/2 radians,

the COS(angle) ratio approaches> 0 and SIN(angle) >1

Trig functions are Unary Operators, they take one argument,

an angle expressed in Radians,

and return to you the ratio of the triangle sides involved.

The COS returns the ratio of the leg adjacent the angle

divided by the hypotenuse. adj/hyp

The SIN function returns the leg opposite the angle fed the function

divided by the hypotenuse. opp/hyp

The third main trig function is TAN, tangent. opp/adj

The TAN(angle in radians) returns

the ratio of the opp side divided by the adjacent side.

In SmallBASIC this is very handy because side opposite/ side adjacent

is the same as y/x the same two items that define a point location!

Can you see what this means?

Does a chill of revelation run up and down your spine?

Oh! that means for any point x,y on the screen,

I can find the angle it is from the origin!

It is the ATAN(y/x)

We are almost ready now to play pin the tail on the donkey! Hang in there...

The A before the TAN, the A before the SIN, COS... means ARC

ATAN is pronounced ARC TANGENT

ACOS is pronounced ARC COSINE

ASIN is pronounced ARC SIN

In each the ARC means The Angle whose Trig Ratio is: (ratio of two sides)

so ATAN is the Angle whose TAN ratio is (opp leg lentgh/adj leg length)

We feed ATAN a leg ratio argument and it tells what angle that ratio came from.

ARC in the Language reference is referred to as "the inverse"

If TAN( an angle) = ratio then

ATAN( ratio=opp/adj ) = an angle (in radians)

DEG(ATAN(opp/adj) = an angle in degrees

You can think or the A in front of TAN or COS or SIN as give me "An Angle"

without the A you get side ratios.

A donkey is mounted on the SmallBasic screen

so that it's tail should be pinned at 0,0 ....

see ATAN for "pin the tail.bas"

shian replied on Permalink

## re: Trig Lesson two

This information is unavailable elsewhere.

Wikipedia gives you formulas that look like ancient Egyptian glyph.

Others, explain what is TAN - but don't even mention a single word about Arc TAN (as if there is absolutely no relation).

This is really a valuable and practical information. (an extra VERY SHORT DEMO for this info would be even more astonishing).

Thanks again.

Edit: by SHORT DEMO I just mean few lines of code that illustrate how it's written in SmallBASiC, for creating the most simple shape... (you know, chunk by chunk... it's really a new world for me and maybe for some others).

MGA replied on Permalink

## TAN use.bas

Like this? It would have been in 3 lines but it seems print usg needs it's own line

TAN used in a short code example:

`' TAN use.bas SmallBASIC 0.12.2 [B+=MGA] 2016-03-14`

for degrees = 0 to 360 step 15

? "For angle (degrees) = ";

? usg "###";degrees;

?" (or ";

? usg "#.00";rad(degrees);

?" radians ) the TANgent is ";

? usg "#####.0000";tan(rad(degrees)) '<== typical to convert degrees to radians (with RAD) before TAN call

next

?:?"As the TANgent approaches 90 or 270 degrees the TANgent becomes 1/0 which is undefined."

pause

I should make clear, TAN is not used directly for drawing on screen but indirectly it helps by calculating angles and lengths which is needed for drawing point to point or step drawing from last drawing x,y point.

shian replied on Permalink

## re: TAN use.bas

Yes, short example like this, or with one single shape on screen, is very very useful.

For instance, I can show an example program of using bitwise XOR in calculating CRC-32 checksum, which is very useful code. But nobody will understand anything, because CRC-32 is complicated algorithm by itself. And it's very hard to understand how XOR is used in this algorithm unless you already very experienced with bit manipulation.

It's useful to give a short and clear examples, something like:

"How can I draw a line from point (x1,y1) to (x2,y2) using TAN."

Later, if I know how to draw one line with TAN, then I will be able to draw a 3D box (maybe..).

Small chunks...

MGA replied on Permalink

## TAN use again

TAN use again in a few lines of code but I have to try and explain what is going on or it would be void of meaning.

`'Short TAN use.bas SmallBASIC 0.12.2 [B+=MGA] 2016-03-14`

'in the following example it is important to keep in mind

'x,y when used in TAN are relative positions to x1,y1 and not

'absolute screen coordinates

'you can move x1,y1 anywhere on screen!!!!

x1=250:y1=150 'x,y of point 1 we will mark with yellow circle

circle x1,y1,2,1,14 'highlight start point in yellow radius 2

'circle x,y,r,aspect,c 'aspect 1 is circle, !1=ellipse

'say you want to make a line 60 degrees from x1,y1 and

'you need the x2 100 greater than x1 or x2 = x1+100

'since we know TAN=y/x (though you need a diagram to see it)

'then TAN(RAD(60)) = y/100

'then 100*TAN(rad(60) = y by algebraic mult 100 both sides

'so y2=y1+100*TAN(rad(60)

line x1,y1,x1+100,y1+100*TAN(RAD(60)) '<== HERE IS TAN

'so here we used TAN to calculate the y change in height

'from y1 to create a 60 angle from point x1,y1 and x2,y2 AND

'AND make it so x2=100 more than x1

'check draw arc at x1,y1 with 100 radius

'for 0 degrees start to 60=2*pi/6 radians end arc, 1=aspect

arc x1,y1,100,0,2*pi/6,1,14 '14=color yellow

`pause`

shian replied on Permalink

## re: TAN use again

SUPREME; BRILLIANT; WELL DOCUMENTED; PRACTICAL; USEFUL; MERCIFUL, KNOCKS OUT.

And the final jump back spinning kick (AKA jump hook kick):

EVEN ME CAN UNDERSTAND IT AND PRACTICALLY USE IT !I could never grasp those holy Aztec keywords... TAN, ASIN... and with H !!! ASINH... Mercy!.

Please continue this way...

Gravely.

--

P.S. sorry for starting a new "comment" - my screen is so small that I cannot see the message by using "reply".

MGA replied on Permalink

## Hi shian

I guess I hit the spot. :))

I drew your cube just by defining 4 points for one square face and then using this code for the back 4 points.

But I didn't stop there. I turned TAN(RAD(60) to TAN(RAD(angle) and put an angle increase in a loop and when...

... well, I spent the night bouncing and spinning and growing/shrinking the cube. ;)

shian replied on Permalink

## re: Hi shian

You certainly hit the spot.

"... well, I spent the night bouncing and spinning and growing/shrinking the cube. ;)"

I know holistic therapist who cured other programmers...

Look, I've made an example program for CIRCLE, using your TAN lesson!

But I only changed the TAN with SIN, and it works....

Now please, repeat that same lesson of TAN - for SIN. So I'll understand why and how it works (just copy the TAN lesson "TAN use again" and update it to explain how SIN works).

Thanks !