Yeah, you're right. sin = opossite leg / hypotenuse The hypotenuse is always the greatest side of the triangle, so: opposite leg < hypotenuse, so: the result of this division (the sine) can never be over 1.
Very well. Of course, in the sine definition you´ve found the answer. The same happens for the cosine function. The greatest value for both is 1. Do you dare to think about the rest of the trigonometric functions?
secant and cosecant = 1/x x = sine or cosine -1<x<1 So it can be any integer except those going from -1 to 1.
tangent = sine / cosine = x / y -1<x<1 -1<y<1 It can be any value with no expeptions. In the 1st quadrant, it must be positive (positive / positive) In the 2nd, negative (neg / pos) 3rd, pos (neg / neg) 4th, neg (pos / neg)
cotangent: 1 / tangent tangent: any value 1 / any value = any value It can be any value, like the tangent. Its signe (positive or negative is the same as the tangent's one.
That´s fine. But it´s much easier to answer using mathematic language. In this case, it´s better to use the term RANGE of a function. So, we´d say: The range of y=sinx is [-1,1] The range of y=cosx is [-1,1] The range of y=tanx is R The range of y=cosecx is R-(-1,1) The range of y=secx is R-(-1,1) The range of y=cotanx is R (Although it´s possible to write the sets using the infinite symbol).
I think that it's not possible. It's easy to explain orally with the unit circle, but explaining it written would take many lines.
ReplyDeleteyou don´t need the unit circle to explain it. Think about the definition of the sine function.
ReplyDeleteYeah, you're right.
ReplyDeletesin = opossite leg / hypotenuse
The hypotenuse is always the greatest side of the triangle, so: opposite leg < hypotenuse, so: the result of this division (the sine) can never be over 1.
Very well. Of course, in the sine definition you´ve found the answer. The same happens for the cosine function. The greatest value for both is 1.
ReplyDeleteDo you dare to think about the rest of the trigonometric functions?
cosine: the same as the sine.
ReplyDeletesecant and cosecant = 1/x
x = sine or cosine
-1<x<1
So it can be any integer except those going from -1 to 1.
tangent = sine / cosine = x / y
-1<x<1 -1<y<1
It can be any value with no expeptions.
In the 1st quadrant, it must be positive (positive / positive)
In the 2nd, negative (neg / pos)
3rd, pos (neg / neg)
4th, neg (pos / neg)
cotangent: 1 / tangent
tangent: any value
1 / any value = any value
It can be any value, like the tangent.
Its signe (positive or negative is the same as the tangent's one.
That´s fine. But it´s much easier to answer using mathematic language. In this case, it´s better to use the term RANGE of a function.
ReplyDeleteSo, we´d say:
The range of y=sinx is [-1,1]
The range of y=cosx is [-1,1]
The range of y=tanx is R
The range of y=cosecx is R-(-1,1)
The range of y=secx is R-(-1,1)
The range of y=cotanx is R
(Although it´s possible to write the sets using the infinite symbol).