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# 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.
- 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.

**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>

**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

**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

**Example 4**- Given angle a = 210
^{o}, find the sin(a) , cos(a) and tan(a).

- Solution
- Since angle a = 210
^{o}= 180^{o}+ 30^{o}, 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 = 30
^{o}. In a right triangle, the length of the leg opposite 30^{o}is one-half of the hypotenuse.

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