I were reading Calculus notes and I found out that when pi is used as an angle of a trigonometric function it's value is 180 degrees and when the same pi is outside that function its value changes to 22/7= 3.14.......aprx
like cos(pi) = x ------> cos(180drgrees) = x
but also cos(pi) = x -------> 3.14.....aproximately = arccos(x)
Why?????
If a right triangle has hypotenuse of length 5 feet and one leg of length 3 feet, what is the length of the other leg? Both "4 feet" and "48 inches" are perfectly valid answers. It is the same thing here: arccos(-1)= 180 degrees or radians are both perfectly valid answers- they just use different units of angle measurement.'
(Since you mention that this question arose in Calculus, you should be aware that "degrees" are almost never used in Calculus and above. "Sine" and "cosine" are used in far more than just "triangle" problems and there are definitions of "sine" and "cosine" that have nothing to do with "angles". In such problems, we, unless there is very good reason to do otherwise, we typically interpret the variable "x" in radians rather than degrees. The basic derivative identities, (sin(x))'= cos(x) and (cos(x))'= -sin(x), are only true if x is in radians.)
(However, it is still NOT correct to say that "is 180 degrees". radians is the same as 180 degrees but is a specific number. I suspect you understood that but I wanted to clarify. And, while I am at it, is close to "22/7" but not equal to it! Of course, you knew that because you also wrote "pi= 3.14...." indicating the infinite decimal expansion.)
actually cos(180°) = cos(π) is wrong on several levels.
first of all, these are two different functions. the one on the left should actually be called cos_{D} (or something similar), as it returns a value based on an input in degrees. this is a totally different function than cosine, but the two ARE related:
cos_{D}(x) = cos(180x/π)
now while 180 degrees = π radians, this does not mean that 180 = π, any more than 3 feet = 1 yards means 3 = 1. degrees and radians are two different units of measurement (believe it or not, there are other units of angular measure as well, such as grad/gradians, sign (30 degrees), and my personal favorite, turn).
the value of π never changes. it's just a number like 5, or √2. the values of cos_{D} and cos are different, because they are different functions (by a "scaling factor" of their argument).
Actually I encountered it in this question
QNo4: If f is differentiable on [a,b] and f-prime (derivative of f) decreases strictly on [a,b] Prove that,
f-prime(b) < {f(b) - f(a)}/{b-a} < f-prime(a)
How this result modifies if f-prime increases strictly on [a,b] Hence show that, if f(x) = sinx , a=pi/3 b=61pi/180 then
sin61degree - sin60degree < 0.0088
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didn't have option to use prime symbol so used f-prime instead for derivative representation hope you guys are okey with it
as i said before (but for cosine),
we have: sin_{D}(x) = sin(180x/π).
for x = a = π/3, 180x/π = (180(π/3))/π = 180/3 = 60.
for x = b = 61π/180, 180x/π = (180(61π/180))/π = 61.
thus sin(a) = sin_{D}(60)
sin(b) = sin_{D}(61).
yes, as units of radial measure. but "sine in degrees" is not the same as "sine in radians". they are different functions. for example, if i tell you f(t) = t^{2}, and you are given t in seconds, the answer is in the same unit (seconds squared). if i tell you t is in feet, f(t) is in "square feet".
the rule: "take the input, and square it" doesn't care about the input "type", you give it a number, it spits out a number.
sin_{D}(π) = 0.054803665148789530887748713539833 (approximately)
sin(π) = 0
so sin_{D}(x) and sin(x) cannot possibly be the same function (they are *similar* though).
every textbook that has written sin(180 degrees) = sin(π radians) is just flat-out wrong. a function is *uniquely* defined by its values on its domain (that's what it *means* to be a function).
one unfortunate consequence of this poor thinking, is that people think somehow π can mean "different things" (which is why this thread even exists). it does not. "sine" is actually what means different things, and unless we want to totally abandon the concept of "function" (and i don't think we do), it needs to be stressed there are two "sine functions" (and we should label them differently, so no one gets confused).
unfortunately, trigonometry started with people using "one kind of sine", and we now realize that "the other kind of sine" makes more sense (for reasons that aren't usually clear until one considers more advanced mathematics than are typically taught at a first exposure to trigonometry).
only holds one value, 3.14159... The reason you might be thinking that is because radians equals 180 degrees. Note that the numbers have units attached to them. It's like saying 12 inches equals one foot, but obviously, 12 does not equal 1.
In trigonometry, we use radians = 180 degrees (a radian is defined as the angle on a circle such that the length of the arc is equal to the length of the radius). Yes, it's still the same 3.1415... number. 22/7 is just an approximation.