sin and cos

[ydoaPs]

alpha is the angle across from side a. beta is the angle across from side b. gamma is the angle across from side c.

[bh_doc]

The inverse you are thinking of is the "multiplicative inverse," which is the same thing as reciprocal. For a nonzero number x, its multiplicative inverse is 1/x. This is different from an inverse function. For a function f(x), its inverse is a function g(x) such that f(g(x)) = 1 for all x.

 

Uh, shouldn't that be f(g(x)) = x for all x?

[Dave]

Indeed it should be.

[Mace Sin]

Hey, I'm a freshman in high school taking Geometry right now, we just finished a segment of a chapter on sines, cosines, and tangents.

 

Oprah/Has = Opposite/Hypotenuse = Sine

 

A/Huge = Adjacent/Hypotenuse = Cosine

 

Old/Asymptote = Opposite/Adjacent = Tangent

 

Ha ha.

 

But, I digress, here's my question:

 

What's the basic way to work out problems pertaining to this area without a calculator? I saw a table, but she (the teacher) would never teach us in class how to use it.

 

(I hate being so dependent on calculators.)

[SilentQ]

Back in the olden times, you would use a book of tables containing sines, cosines, tangents, arcsines, arccosines, arctangents, natural logarithms and probably natural antilogarithms (exponentials). These were calculated by hand by a person called (confusingly) a computer. The computer would use some kind of infinite series (a Maclaurin or Taylor expansion or similar).

 

It is also possible to build a mechanical device for calculating the sine. I'm not aware of a similar device for the cosine and tangent, but one could no doubt be invented with a little ingenuity.

[Mace Sin]

Back in the olden times' date=' you would use a book of tables containing sines, cosines, tangents, arcsines, arccosines, arctangents, natural logarithms and probably natural antilogarithms (exponentials). These were calculated by hand by a person called (confusingly) a computer. The computer would use some kind of infinite series (a Maclaurin or Taylor expansion or similar).

 

It is also possible to build a mechanical device for calculating the sine. I'm not aware of a similar device for the cosine and tangent, but one could no doubt be invented with a little ingenuity.[/quote']

 

So, would I be required in higher education to find it by hand?

 

I'm worried that in the future classes (as in my college years) will not be so heavily dependent on machines, but rather by your own skill and intellect.

[SilentQ]

If anything, I've got more dependent on machines as I've progressed. I was told to acquire a graphical calculator upon entering the sixth form (last two years of high school in Britain). I have certainly never had to use infinite series to calculate trig functions. The Maclaurin expansion is used to prove de Moivre's Theorem, which lies at the basis of complex analysis, but using it to calculate values of functions is counterproductive when you can use a calculator.

 

There are certain values you might want to commit to memory. In radians they are the following.

 

[math]\sin 0=\cos\frac{\pi}{2}=0[/math]

 

[math]\sin\frac{\pi}{6}=\cos\frac{\pi}{3}=\frac{1}{2}[/math]

 

[math]\sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}[/math]

 

[math]\sin\frac{\pi}{3}=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}[/math]

 

[math]\sin\frac{\pi}{2}=\cos 0=1[/math]

 

[math]\tan 0=0[/math]

 

[math]\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}[/math]

 

[math]\tan\frac{\pi}{4}=1[/math]

 

[math]\tan\frac{\pi}{3}=\sqrt{3}[/math]

 

[math]\tan\frac{\pi}{2}=\infty[/math]

 

Memorising these will certainly serve you well if you intend to take further mathematical study. If you prefer the angles in degrees, 180 degrees are equal to [math]\frac{\pi}{2}[/math] radians.

 

If it helps, think of a right-angled isosceles triangle for [math]\frac{\pi}{4}[/math], and an equilateral triangle cut in half along a line of symmetry for the rest. It is the work of ten seconds to draw a triangle in the margin of an exam script if necessary.

[cosine]

Hey' date=' I'm a freshman in high school taking Geometry right now, we just finished a segment of a chapter on sines, cosines, and tangents.

 

[b']Oprah/Has[/b] = Opposite/Hypotenuse = Sine

 

A/Huge = Adjacent/Hypotenuse = Cosine

 

Old/Asymptote = Opposite/Adjacent = Tangent

 

Ha ha.

 

The pneumonic I've found most interesting is:

 

Some Old Hippie

Caught Another Hippie

Tripping On Acid.

[The Thing]

In my school, all the math teachers use the mnemonic SOH CAH TOA. It's so ridiculous unimagniative and terrible-sounding that everybody remembers it. Meh, whatever floats your boat.

 

Trig ratios for special angles are very easy to memorize (even if you don't know radians) and if you can't memorize you can draw a triangle on the spot. The special angles include 0, 30, 45, 60, 90, all angles in other quadrants with these reference angles, and all coterminal angles. All you need to know is the sides of a 45-45 right triangle (1,1, and root 2) and a 30-60 right triangle (1, 2, and root 3), and how to draw a unit circle. You should know these fine. Then just know the +/- of each trig ratio in each quadrant and you're set for special angles without calculators.

[SilentQ]

Beg pardon. 180 degrees is [math]\pi[/math] radians, not [math]\frac{\pi}{2}[/math] as I said before. Sorry, my bad.

[The Thing]

Degrees to Radians: [math]\frac{DEG* \pi}{180}[/math]

Likewise, Radians to Degrees: [math]\frac{RAD*180}{\pi}[/math]