Exploring Mathematics without a Calculator: A New Endeavor

[kai0ty]

ive started an endeavor (being an amature mathematician) to find sin cos and tan w/o the use of a calculator. i was wondering if this had already been done because it will save me some time. anyone know how?

 

[jeffceth]

Originally posted by kai0ty
ive started an endeavor (being an amature mathematician) to find sin cos and tan w/o the use of a calculator. i was wondering if this had already been done because it will save me some time. anyone know how?

I suppose that you could diagram it geometrically and approximate calculations by hand to a closer and closer degree of accuracy.


sincerely,
jeffceth

 

[lastlaugh]

The trick is really to just find sin, then use relationships to figure out the other 2.

 

[Guybrush Threepwood]

why don't you use MacLaurin series ?

 

[Hurkyl]

A common thing before calculators was to consult a book of tables when you wanted to find the value of a trig function.

 

[jcsd]

The tangent, cosine and sine functions are what as known as transcendental funtions which means basically they cannot be written as polynomials. A calculator works them out using series like those that Guybrush linked to (though they do have several important identities for certain values of cos x, sin x and tan x pre-stored in their memory) and works then out to the nth decimal place.

As Hurkyl said in the days before calculators you would have a book of tables which would list these functions for several values of x, infact some books still have tables at the back for logorithmic functions (which are another example of trancendental functions).

 

[kai0ty]

that series is what i was looking for actually. i know about the tables and how the calculator works and stuff, i just thought it would be cool to do it w/o a calculator because my math teachers said it was impossible...

 

[kai0ty]

um bit of a problem w/ that series, there wrong. i entered them in a calculator and the answeres were off noticably? is there some special way to read those that i don't know of being only in algebra 2, or is he just wrong? the thing i did was replace x with a number i chose like 2. when i entered it in i got something a lot different than when i put it in my caclulator just as sin(2) is that wrong?

 

[Lonewolf]

Is your calculator working in degrees, by any chance? x needs to be in radians.

 

[kai0ty]

muhaha there you go. why was it in degrees i wonder... o well thanks a lot.

 

[jcsd]

Originally posted by kai0ty
um bit of a problem w/ that series, there wrong. i entered them in a calculator and the answeres were off noticably? is there some special way to read those that i don't know of being only in algebra 2, or is he just wrong? the thing i did was replace x with a number i chose like 2. when i entered it in i got something a lot different than when i put it in my caclulator just as sin(2) is that wrong?

Yes x is in radians, but also note that using those series the answer is always going to be a little off the real value as these are infinite series.

 

[Integral]

When using a few terms of the infinite series your error can be found by simply looking at the magnitude of the first term you did NOT calculate.

Notice that if your angle is small, that is less then about .2 radian you have

x= sin(x) and x = tan(x) This is called the small angle approximation and is commonly used in Physics. For example the formula for the the period of a pendulum is the result of a small angle approximation. Which means of course the formula will break down if the swings of your pendulum are to big.

 

[kai0ty]

yea as the number got bigger it got more accurtate too. thanks all

 

[StarkyDee]

sup everyone...

since we are on the topic of tan,sin, and cos.. i am in trig right now and understand most of the math. but i don't really understand what tan, sin and cos really are?? tan is a point on the circle where a line touchs the edge? i have made graphs of the sin and cos functio ns, but what do they represent? i really don't understand the philosophy or signif. of these symbols..can someone set me straight please? thanks

 

[chroot]

Originally posted by StarkyDee
since we are on the topic of tan,sin, and cos.. i am in trig right now and understand most of the math. but i don't really understand what tan, sin and cos really are?? tan is a point on the circle where a line touchs the edge? i have made graphs of the sin and cos functio ns, but what do they represent? i really don't understand the philosophy or signif. of these symbols..can someone set me straight please? thanks

Take a unit circle on a normal Cartesian plane. A unit circle has a radius of one unit. Place the circle on the plane so that the circle's center is on the origin of the coordinate system.

Now, consider an angle $$\inline{\theta}$$ that is measured counterclockwise from the positive x-axis. In other words, the point on the circle $$\inline{(1, 0)}$$ is assigned angle 0. As you go around the circle counterclockwise from that point, the angle increases.

The sine function $$\inline{\sin ( \theta )}$$ is defined as the y-axis coordinate of the point on the circle with angle $$\inline{\theta}$$. The cosine function $$\inline{\cos ( \theta )}$$ is defined as the x-axis coordinate of the point on the circle with angle $$\inline{\theta}$$.

The tangent function $$\inline{ \tan ( \theta )}$$ is defined very simply as

$$\tan (\theta) \equiv \frac{\sin{\theta}}{\cos{\theta}}$$

Does this make sense?

- Warren

 

[StarkyDee]

yes that does help me out Warren, thanks. so from the perspective of a 2d graph: (cos,sin) is (1,0) -in your example. so i would guess arcsin,cos,tan are just inverses: such as cot = cos/sin.. but i don't understand why you you would need cot,csc,sec?

 

[chroot]

Originally posted by StarkyDee
yes that does help me out Warren, thanks. so from the perspective of a 2d graph: (cos,sin) is (1,0) -in your example.

Precisely.

so i would guess arcsin,cos,tan are just inverses: such as cot = cos/sin..

Not quite. Arcsin, arccos and arctan are indeed inverse functions. Arcsin, for example, returns the angle of a given y-coordinate on the unit circle. The cotangent is not an inverse. It's just the reciprocal of the tangent.

but i don't understand why you you would need cot,csc,sec?

You don't "need" them. They are just different names for the reciprocals of tan, sin, and cos, respectively.

- Warren