The Definition of Trigonometric Functions of Any Angles

The Definition of Any Angles of Trigonometric Functions
Let a is an any angle, P ( x , y ) is a point in terminal side of angle a. Its horizontal coordinate is x and vertical coordinate is y.
the graph of the definition of the trogonometry definition
the definitopn of the trigonometry functions
Note: 1. When the terminal side of angle a coincides with x-axis, y = 0, then cot a and csc a are undefined. 2. When the terminal side of angle a coincides with y-axis, x = 0, then tan a and sec a are undefined. 3. The value of trigonometric function does not change when the P ( x , y ) move along the terminal side of angle a.
The Domain of Trigonometric Functions and its Value
Angle a is a variable of trigonometric functions which has an unit of radian. P(x, y) is a point on terminal side of angle a. When x = 0, the terminal side of the angle a coincides with y-axis, then the value of the set of a is { a | a = pi/2 + k pi, in which k is an integer}, When y = 0, the terminal side of the angle a coincides with x-axis, then the value of the set of a is { a | a = k pi, in which k is an integer}. The value of trigonometric function is a ratio, the function is undefined when the denominator of the ratio is zero. The domain is a set of value that make the denominator of the ratio of trigonometric function is not zero.
Trigonometry function Domain Value
sin a any real number |sin a| <= 1
cos a any eal number |cos a| <= 1
tan a a not equal to kpi + pi/2 any real number
k = ...-3,-2,-1,0,1,2,3,4,5 ...
Example 1
Let P ( -4, 3 ) be a point on the terminal side of angle a. Find the sin(a) , cos(a) and tan(a).
Solution>
The graph of the trigonometry function definition example1
The solution of the trigonometry function example1
Example 2
Let P ( -4, -3 ) be a point on the terminal side of angle a. Find the sin(a) , cos(a) and tan(a).
Solution
The grapg of the trigonometry function definition example2
The solution of the trigonometry function definition example2
Example 3
Let P ( 4, -3 ) be a point on the terminal side of angle a. Find the sin(a) , cos(a) and tan(a).
Solution
The graph of the trigonometry function example3
The solution of the trigonometry function example3
Example 4
Given angle a = 210o, find the sin(a) , cos(a) and tan(a).
Solution
Since angle a = 210o = 180o + 30o, so the terminal side of angle a lies in Quadrant III . So  sin a < 0 , cos a < 0, and tan a > 0.  See the figure below. TP = y  and y < 0 . TO = x  and x < 0
Note: the angle TOP = 30o. In a right triangle, the length of the leg opposite 30o is one-half of the hypotenuse.
Trigonometry function definition example4
The given condition of the trigonometry function example4
The solution of the trigonometry function example4