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Definition of the Six Trigonometric Functions
We begin with a unit circle, a circle of radius 1. Each point on the circle defines an angle
with the positive axis and has an and a coordinate. We make the following definitions.
These definitions lead immediately, via the Pythagorean Theorem, to the Pythagorean Identities:
STEP 2005, Math I, #7: The notation
denotes the product .
Simplify the following products as far as possible:
(i)
(ii)
(iii)
where
is even.
STEP 2012, Math III, #5: (i) The point with coordinates , where and are rational numbers,
is called:
an integer rational point if both
and are integers;
a non-integer rational point if neither
nor is an integer.
(a) Write down an integer rational point and a non-integer rational point on the circle .
(b) Write down an integer rational point on the circle . Simplify
and hence obtain a non-integer rational point on the circle .
(ii) The point with coordinates where , , and are rational numbers,
is called:
an integer 2-rational point if all of
, , and are integers;
a non-integer 2-rational point if none of
, , and is an integer.
(a) Write down an integer 2-rational point, and obtain a non-integer 2-rational point, on the circle .
(b) Obtain a non-integer 2-rational point on the circle .
(c) Obtain a non-integer 2-rational point on the hyperbola .
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