 Hello and welcome to the session. In this session first we will discuss about angles. Basically an angle is a measure of rotation of a given ray about its initial point. This OA is the original ray which is called the initial side and after rotation we get the final position of the ray which is OB. This is called the terminal side and the point of rotation which is OA is called the vertex. If the direction of the rotation is anti-clockwise then we say we get a positive angle and if the direction of rotation is clockwise we get a negative angle. We can also say the measure of an angle is the amount of rotation performed to get the terminal side from the initial side. Next we shall study degree measure which is another important unit of measurement of an angle. If the rotation from the initial side to terminal side is 1 upon 360th of a revolution the angle is said to have a measure of 1 degree which is written as this. Next we have radiant measure which is also an important unit of measurement of angle. Angles subtended at the center by an arc of length one unit in a unit circle which is a circle of radius one unit is said to have a measure of one radian. We also say that one complete revolution of the initial side subtends an angle of 2 pi radian in a circle of radius r an arc of length l subtends an angle theta radian at the center we have theta is equal to l upon r or we can say that l is equal to r theta. Radiant measures and real numbers are considered as one and the same. Next we discuss relation between degree and radian we have 2 pi radian is equal to 360 degrees or we can say that pi radian is equal to 180 degrees. We also say that radian measure is equal to pi upon 180 multiplied by degree measure also degree measure is equal to 180 upon pi multiplied by radian measure. Let's try and express the angle 36 degree in radian measure. Its radian measure would be given by 36 degrees is equal to pi upon 180 multiplied by 36 radians which comes out to be equal to pi upon 5 radians that is we have 36 degrees is equal to pi upon 5 radians. Next we discuss trigonometric functions we have sin x equal to 0 implies that x is equal to n pi where n is any integer then we have cos x equal to 0 implies that x is equal to 2n plus 1 multiplied by pi by 2 where again n is any integer then we have some other trigonometric functions like cos x equal to 1 upon sin x where x is not equal to n pi where n is any integer then sec x equal to 1 upon cos x where we have x is not equal to 2n plus 1 multiplied by pi by 2 for any integer n then tan x is equal to sin x upon cos x where again x is not equal to 2n plus 1 multiplied by pi by 2 for any integer n and cos x is equal to cos x upon sin x where x is not equal to n pi for some integer n. We also have for all real x sin square x plus cos square x is equal to 1 1 plus tan square x is equal to sec square x and 1 plus cos square x is equal to cos x square x. Now we discuss sin of trigonometric functions this table shows the signs of the trigonometric functions in different quadrants like in the first quadrants all the trigonometric functions sin x cos x tan x cos x sin x and cos x are positive in the second quadrant only sin x and cos x are positive all other are negative in the third quadrant tan x and cot x are positive rest all are negative and in the fourth quadrant cos x and sec x are positive rest all are negative. Next we discuss domain and range of trigonometric functions from the definition of sin and cosine functions we observe that they are all defined for all real numbers and we also have that sin x is greater than equal to minus 1 and less than equal to 1 and cos x is greater than equal to minus 1 and less than equal to 1. Then for the functions y equal to sin x and y equal to cos x domain is the set of all real numbers and range is the closed interval minus 1 1 where we have y is greater than equal to minus 1 and less than equal to 1 for the function y equal to cos x that is 1 upon sin x we have domain is the set x such that x belongs to r and x is not equal to n pi where n belongs to z and range is the set y such that y belongs to r y is greater than equal to 1 or y is less than equal to minus 1 then we have y equal to sec x that is 1 upon cos x for this we have domain is the set x such that x belongs to r and x is not equal to 2n plus 1 multiplied by pi by 2 where n belongs to z and range of this function is the set y such that y belongs to r y is less than equal to minus 1 or y is greater than equal to 1 then we have function y equal to tan x domain of this function is the set x such that x belongs to r and x is not equal to 2n plus 1 multiplied by pi by 2 where again n belongs to z and range is the set of all real numbers the next trigonometric function is y equal to cot x domain for this function is given by the set x such that x belongs to r and x is not equal to n pi where n belongs to z and range is the set of all real numbers this table shows the behavior of the trigonometric functions in different quadrants like in the first quadrant when x increases from 0 to pi by 2 sin x increases from 0 to 1 cos x decreases from 1 to 0 tan x increases from 0 to infinity cot x decreases from infinity to 0 sec x increases from 1 to infinity cos x decreases from infinity to 1 then in the second quadrant when x increases from pi by 2 to pi sin x decreases from 1 to 0 cos x decreases from 0 to minus 1 tan x increases from minus infinity to 0 cot x decreases from 0 to minus infinity sec x increases from minus infinity to minus 1 and cos x increases from 1 to infinity in the third quadrant when x increases from pi to 3 pi by 2, we have sin x decreases from 0 to minus 1, cos x increases from minus 1 to 0, tan x increases from 0 to infinity, cot x decreases from infinity to 0, second x decreases from minus 1 to minus infinity, and cos x increases from minus infinity to minus 1. Then in the fourth quadrant, when x increases from 3 pi by 2 to 2 pi, sin x increases from minus 1 to 0, cos x increases from 0 to 1, tan x increases from minus infinity to 0, cot x decreases from 0 to minus infinity, second x decreases from infinity to 1, and cos x decreases from minus 1 to minus infinity. Let's take sec theta equal to minus 13 upon 12, and we have that theta lies in the second quadrant. Let's try and find out the values of the trigonometric functions. We have sec theta equal to minus 13 upon 12, and we know that cos theta is equal to 1 upon sec theta, so we get cos theta is equal to minus 12 upon 13, and we know that tan square theta is equal to sec square theta minus 1, this is equal to minus 13 upon 12, we hold square minus 1, so from here we get tan square theta is equal to 25 upon 144, hence we have tan theta is equal to plus minus 5 upon 12, now since we have theta is in the second quadrant, so we will take the negative value of tan theta, so we get tan theta is equal to minus 5 upon 12, then we have cot theta is equal to 1 upon tan theta, so we get cot theta equal to minus 12 upon 5, now sin theta is equal to tan theta multiplied by cos theta which is equal to minus 5 upon 12 multiplied by minus 12 upon 13 which gives the value equal to 5 upon 13, so we have sin theta is equal to 5 upon 13, and cos theta is equal to 1 upon sin theta, thus we get cos theta is equal to 13 upon 5, so this is how we find the trigonometric functions, this completes the session, hope you have understood the concept of angles trigonometric functions.