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5.2 and 5.3 Right Angle Trigonometry
1. Trigonometric Functions of Acute Angles
4. Fundamental Identities

Reciprocal Identities

Quotient Identities

Pythagorean Identities
4. Complementary Angle Theorem
Cofunctions of complementary angles are equal
5. Exact values of Trigonometric functions for some special angles
5.4
So far we have learned the definitions and applications of the six trigonometric
functions for acute angles. Now we extend these definitions to angles of any
measure.
1.
Note:

P=(x,y) can be any point on the terminal side of θ except for the origin (0,0).

In the definition above r will always be positive since it is a distance.

x can be either positive or negative and the same is true y as well.

Therefore,
2. Trig functions for quadrantal angles.
What is a quadrantal angles? They are angles where the terminal side falls on
either one of the axes. These angles DO NOT form triangles with the x-axis.
To find values of all the trig functions for quadrantal angles we pick a point on
the terminal side and then employ the formulae for the definitions of
trigonometric function for any angles.
3. Coterminal angles to find the exact value of a trigonometric function.
Two angles in standard position are said to be coterminal if they have the
same terminal side.
4. Exact values of trigonometric function using reference angles
What is reference angle of a general angle?
Let θ be a nonacute angle in standard position that lies in a quadrant. Its
reference angle is the positive acute angle formed by the terminal side of θ
and the x-axis.
To find trigonometric functions of angles using reference angles we use the
following formulae:
5.5
Unit Circle Approach
The length of the intercepted arc is t. This is also the radian measure of the
central angle. Thus, in a unit circle, the radian measure of the central angle is
equal to the length of the intercepted arc. Both are given by the same real
number t.
Doman and Range of Trigonometric functions
Properties of trig functions
A) Periodicity Property
The trigonometric functions are periodic, that is, there is regular repetition of the
values of the functions over a certain interval of angle.
The sine and cosine functions are periodic with 2Π i.e.
sin( +2Πk) = sin( ) and cos( +2Πk) = cos( ) where k is any integer.
Therefore,
csc( +2Πk) = csc( ) and sec( +2Πk) = sec( ) where k is any integer.
The tangent function is periodic with Π i.e.
tan( +Π) = tan( ) where k is any integer.
Therefore,
cot( +Π) = cot( ) where k is any integer.
B) Even and Odd Properties
P=(x,y)
Q=(x,-y)