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Unit
Circle Definition of Sine and Cosine Functions
The trigonometric functions can be
defined in terms of a unit circle, i.e. a
circle of radius one.
The
sin/cos Triangle
If the unit circle is placed at the
origin of a rectangular coordinate system with the angle q measured from
the positive x-axis to the terminal side, then the point on the unit
circle where the terminal side intersects the unit circle is defined to
be (cos q, sin q), i.e. the first coordinate of a point on the unit
circle is the cos q and the second coordinate is sin q.
The
tan/sec Triangle
The tan q and sec q are defined
by a triangle whose height is tangent to the unit circle at the point
(1, 0) and whose hypotenuse is on the terminal side of the angle.
The
cot/csc Triangle
The cot q and csc q are defined
by a triangle whose height is one and whose hypotenuse is on the
terminal side of the angle.
The
sin/cos, tan/sec, and cot/csc Triangles
All three of the
triangles used to define the trigonometric functions are shown in figure
below.
Using the sin/cos,
tan/sec, and cot/csc Triangles to Establish Basic
Trigonometric Identites
The three similar triangles sin/cos,
tan/sec, and cot/csc are extracted from the figure. A fourth similar
triangle is shown with the adjacent, opposite, and hypotenuse sides
labelled.
The definition of the six
trigonometric functions and other useful identities follow from using
the fact that the ratio of corresponding sides of similar triangles must
be the equal. The results are:
Using the sin/cos,
tan/sec, and cot/csc Triangles to Determine the Pythagorean
Idenities
The Pythagorean Theorem states: in any
right-angled triangle, the sum of the squares of the lengths of the
sides containing the right angle is equal to the square of the
hypothenuse. In short c2 = a2 + b2.
Applying the Pythagorean
Theorem to the sin/cos, tan/sec, and cot/csc triangles gives:
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