What does your calculator do when you press the sin/cos/tan button?

Apparently TI calculators use the CORIDC algorithm which involves rotation on a complex plane using complex numbers.

COordinate Rotation DIgital Computer

aka

Voldic’s algorithm

I would think that at least some calculators use (or used) the Taylor Series for the functions.

They would be something a calculator can do fairly easily, as opposed to the sine function itself. Taylor Series use polynomials.

That is more likely something you would see as a mathematics/physics student at the undergraduate level.

You would learn about the sine function being an ‘odd’ function and the cosine function being an ‘even’ function.

Each is an alternating series that starts with a positive term.

If you use more terms, you get more accuracy, but a calculator displays a limited number of terms. So a fairly small number of terms in the Taylor Series will give you a decent approximation for many things.

Also, these Taylor Series are more accurate with smaller values of x using less terms. If you use x = 0, they’re exactly right using only the first term.

SOHCAHTOA CHOSHACAO

We started by looking at evaluating trig functions of angles. They involved 30 60 90 triangles and 45 45 90. So we began by looking at those triangles.

It’s easy enough to derive the ratios for the isosceles triangle.

Then we compared that to the other triangle.

Used SOHCAHTOA and CHOSHACAO.

Often, it helps to draw the angles on the xy axes starting from the 0° position (usually East). Then connect the angle to the x-axis to make a triangle.

Also, the functions are pronounced ‘sine’, ‘cosine’, ‘tangent’. If you’re going to say them out loud, say those, not “sin”, “cos”.

If you get to “csc”, how would you even say that out loud?

“Why doesn’t my calculator shows sin 60 as root three by two when it is in radian mode?”

Saw this question today,

60_degrees_60_radians

If you take the trig function of two different values, the results could be equal, but there is no reason they have to be. In this case, the two results are not equal.

Why is Tan(90°) Wrong? (Undefined)

I’ve drawn a triangle that has an angle close to 90°, it’s looks more like 85°. That is to make it look like a triangle rather than a degenerate triangle (straight line). But I have used the values of the sides for when the triangle has a ninety degree angle.

tan_90

As you go from 85° to 90°, the adjacent side goes from being small to being zero.

Using SohCahToa, you can get the tangent of theta by dividing the opposite by the adjacent side.

1/0

Which is either undefined (in earlier classes) or something like ∞. The limit from the right and left changes the sign though.

What is csc(225°)?

csc(225°)

The cosecant function is the reciprocal of the sine function.

So if you think about the ratios of a triangle for sine, you have opposite/hypotenuse.

For cosecant, it will be hypotenuse/opposite.

Placing the angle on the xy axes

225° is 45° past 180°.

csc_225

The hypotenuse for a unit circle is positive 1. And the opposite side is negative and a value of -1/√2

So you want to calculate 1/(-1/√2) = -√2

“Why is sin(70°) the same as sin(110°)?

From Quora,

Here’s a diagram.

Each is 20°away from 90°. If you think in terms of the ratios of the sides of the triangles for the angles, SOHCAHTOA, then the opposite sides are the same and the hypotenuse is the same. So the ratio is the same for each.

sin_70_sin_110

“How will you evaluate the approximate value of tan 46 degrees?”

Came up with a bit of a strange method for an approximation of tan(46 degrees)

I would first note that 46 degrees is close to 45 degrees.

The tan(45 degrees) = 1 since it’s a 45 degree triangle and the opposite over the adjacent is going to give you 1.

46 degrees is slightly above 45 degrees, so the opposite side is going to be a bit bigger than the adjacent side.

Therefore it will be a bit over 1.

From there I’ll think about how much different it will be than tangent of 45 degrees.

Since you’re given that 1 degree 0.01745 radians,

I think about the opposite side getting a bit bigger and the adjacent side of the triangle (for the 46 degree right triangle) getting a bit smaller (equal amounts).

I call the change ∆. If we call the opposite y and the adjacent x, then the slightly changed sides would change by this amount ∆. And because tanx ~ x for small angles, I’m thinking about the change in the angle 46-45 = 1 degree as the angle, which is fairly small.

So for an approximation, something that I came up with that seems to work…..

tan 46 degrees

 

tan 46 degrees more detail

tan(46 degrees) ~ (1+0.01745)/ (1-0.01745)

= 1.035519821

If I just type in tan(46 degrees), I get

=1.035530314

If I use more accuracy in the calculator for the radian equivalent of 1 degree, I get

tan(46)~ 1.035526642

Which is different by 3.67188 x 10^-6

This method seems to work even for larger changes in angle. For example

tan(50) ~  (1+0.01745*5)/ (1-0.01745*5)

= 1.191180498

On the calculator

tan(50) = 1.191753….

So a bit less accurate, but still accurate to the nearest hundredth.

Not quite sure why this works.

“If I am given a cosine function, how do I write an equivalent sine function without graphing?”

Check out this comparison of sinx and cosx. If you shift sinx to the left by (1/2)π, it is the same as cosx.

sinx_cosx_phase_displacement

You can add a phase displacement by adding or subtracting from the argument of the trigonometric function.

“How to find the height of an equilateral triangle?”

Draw an additional vertical line to split the equilateral triangle into two 30-60-90 triangles. Then you can either use trig functions or the Pythagorean Theorem.

area of equilateral triangle

“How were sine, cosine, and tangent derived?” Discovery, without using calculator

The ancient Babylonians and Egyptians knew some about trigonometry. As did the ancient Greeks.

The Greeks, Euclid and Archimedes, had the law of cosines and law of sines in their work.

Often thought of, it seems, in terms of chords of triangles.

Here’s an example,

chords trigonometry

I saw that Hipparchus of Nicaea did quite a bit with early trigonometry.

You can measure lengths of triangles within a circle using an instrument to measure distance. They could make tables of values.

Later on, with angles, you can still calculate sine, cosine, and tangent of angles using the ratios of triangles.

For a right triangle,

sinθ = O/H

cosθ = A/H

tanθ = O/A

opposite, adjacent, hypotenuse

SOHCAHTOA