Why do scientific calculators even have a cosine key?

Cosine, sine, and tangent are the three properties taught in trigonometry. If we didn't have all three, say, only sine, then calculating things would become extremely complicated. Here are some examples (these are all right triangles):

1) Base angle: 37°, adjacent: 3, find hypotenuse

So we know: sin(37°) = o/h --> h = o / sin(37°) <-- need to find o.

But we know a relationship between o and h:

a² + o² = h² --> o² = h² - a² = h² - 3² = h² - 9

-->

sin(37°) = o/h = √(h² - 9) / h

--> now solve for h

sin²(37)h² = h² - 9 --> h²(1 - sin²(37)) = 9 --> h = 3 / √(1 - sin²(37°))

...which of course you recognize as h = 3 / cos(37°)

There is one other way, you can use your trick:

cos(x) = sin(90 - x)

--> so really we want the cosine:

cos(37°) = a/h = sin(90° - 37°) --> h = 3/sin(53°) <-- which you again recognize as cos(37°)

2) angle: x, adjacent: a, find opposite

Here you want to use: tan(x) = o/a --> o = a * tan(x)

But you either have to do Pythagorean Theorem:

h = √(a² + o²) --> sin(x) = o / √(a² + o²)

-->

o = a * sin(x) / √(1 - sin²(x)) --> which you recognize as o = a * sin(x) / cos(x) = a * tan(x)

Or you again, have to do your trick, but now to do two pre-steps:

o = a * tan(x) = a * sin(x) / cos(x) = a * sin(x) / sin(90 - x)

...instead of just using the tangent which is what you wanted to do all along.

The secants, cosecants, and cotangents, are much simpler and they all work the same way with their respective analogues (just one over). So in the following equation which do you prefer:

3) angle: x, opposite: o, find hypotenuse

sin(x) = o/h --> h = o / sin(x)

or

csc(x) = h/o --> h = o * csc(x)

It's more or less the same steps (in terms of putting it into your calculator).

You should have at least three of them, choose either sine, cosine, and tangent, or cosecant, secant, and cotangent. The choice as to which group you choose to put on the calculator is arbitrary, but I imagine cosine, sine, and tangent are put there because those are the canonical ones we teach in early trigonometry (you usually don't hear about cosecant, secant, and cotangent until later, usually in pre-cal or Algebra II).

The reciprocal functions are not used anywhere near as much as the standard sine/cosine/tangent functions (and in the case they are used, they usually can easily be converted into the standard functions), and so having three extra buttons for them on a calculator isn't really necessary.

Also, with your "since you can just subtract from pi/2 or 90 or whatever, isn't the cosine key completely unnecessary?" question, I believe you are referring the fact that:

cos(x) = sin(π/2 - x),

and so a cosine button isn't necessary since you can just take the complement of the angle you want to take the cosine of, subtract the result from π/2, and then take the sine of it; however, the same thing could be said about the sine button. In the case of sine and cosine, having one button but not the other would be kind of silly.

It's because of how often cosine is used as a separate operation. The other three functions are not used nearly as often.

Another example is logarithms. Most scientific calculators provide log (base 10) and ln (base e). They don't usually provide an arbitrary "base b" operation because any base can be computed by dividing two logs of any other base. This means that only one logarithm base is necessary, but because base 10 and base e are so commonly used, both are typically provided, and other bases would require extra steps.

And by the way, tangent can be computed from sine as well, since you can just divide sine by cosine to get that and you know how to get cosine from sine. But why would you want to?

Why not? You might not need it but some people will need it with their problems. You'll notice that the cosine key is actually useful later when you get to higher maths.

Can someone answer my question?

I don't work for TI, but using this logic, calculators don't need a subtraction or division key either. I think a calculator is a product aimed at the masses, not only to people who already know advanced math.

thats nice but some people dont know that..

why do calculators have a square key because that can be done by simply multiplication of the same number..