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1 Top Concepts Class XI: Maths Ch : Trigonometric Function Chapter Notes. An angle is a measure of rotation of a given ray about its initial point. The original ray is called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex.. If the direction of the rotation is anticlockwise, the angle is said to be positive and if the direction of the rotation is clockwise, then the angle is negative. Positive Angle- Anticlockwise Negative Angle- Clockwise. If a rotation from the initial side to terminal side is 60 of a revolution, the angle is said to have a measure of one degree, I t is denoted by o. 4. A degree is divided into 60 minutes, and a minute is divided into 60 seconds. One sixtieth of a degree is called a minute, written as, and one sixtieth of a minute is called a second, written as th

Thus, o = 60, = 60 5. Angle subtended at the centre by an arc of length unit in a unit circle is said to have a measure of radian 6.

y-coordinate of the point, and the cosine of the angle is the x-coordinate of that point. 7.

2 Thus, o = 60, = Angle subtended at the centre by an arc of length unit in a unit circle is said to have a measure of radian 6. If a point on the unit circle is on the terminal side of an angle in standard position, then the sine of such an angle is simply the y-coordinate of the point, and the cosine of the angle is the x-coordinate of that point. 7. All the angles which are integral multiples of are called quadrantal angles.values of quadrantal angles are as follows: cos 0,sin 0 0 cos 0,sin cos,sin cos 0,sin cos,sin 0

3 8.Cosine is even and sine is odd function cos(-x) = cos x sin(-x) = - sin x 9. Signs of Trigonometric functions in various quadrants In quadrant I, all the trigonometric functions are positive. In quadrant II, only sine is positive. In quadrant III, only tan is positive, quadrant IV, only cosine function is positive. This is depicted as follows 0. In quadrants where Y-axis is positive (i.e. I and II), sine is positive and in quadrants where X-axis is positive (i.e. I and IV), cosine is positive. A function f is said to be a periodic function if there exists a real number T>0, such that f(x + T) = f(x) for all x. This T is the period of function.. sin (+ x ) = sin x so the period of sine is. Period of its reciprocal is also. cos (+ x) = cos x so the period of cos is. Period of its reciprocal is also 4. tan (+ x) = tan x Period of tangent and cotangent function is 5.The graph of cos x can be obtained by shifting the sin function by the factor 6. The tan function differs from the previous two functions in two ways (i)tan is not defined at the odd multiples of / (ii) tan function is not bounded. 7. Function Period y=sin x y=sin (ax) a y=cos x y=cos (ax) a y=cos x y=sin 5x 5

4 8. For a function of the form y= kf(ax+b) range will be k times the range of function x, where k is any real number if f(x)= sine or cosine function range will be equal to R-[-k, k] if function is

4 4 8. For a function of the form y= kf(ax+b) range will be k times the range of function x, where k is any real number if f(x)= sine or cosine function range will be equal to R-[-k, k] if function is of the form sec x or cosec x, Period is equal to the period of function f by a. The position of the graph is b units to the right/left of y=f(x) depending on whether b>0 or b<0 9. The solutions of a trigonometric equation for which 0 x are called principal solutions. 0.The expression involving integer n which gives all solutions of a trigonometric equation is called the general solution.. The numerical smallest value of the angle (in degree or radian) satisfying a given trigonometric equation is called the Principal Value. If there are two values, one positive and the other negative, which are numerically equal, then the positive value is taken as the Principal value. Top Formulae o. radian = 80 o 57 6' approximately. o = radians radians approximately o 80. s= r θ Length of arc= radius angle in radian This relation can only be used when is in radians

5 5 4. Radian measure= Degree measure Degree measure = 80 Radian measure 6. Trigonometric functions in terms of sine and cosine cosec x,x n,where n is any int eger sinx sec x,x ( n ),where n is any int eger cos x sinx tanx,x (n ),where n is any int eger cosx cot x,x n,where n is any int eger tanx 7. Fundamental Trigonometric Identities sin x + cos x = + tan x = sec x + cot x = cosec x 8 Values of Trigonometric ratios: 0 sin 0 cos tan π π π not defined 0 not defined 0 9. Domain and range of various trigonometric functions: Function Domain Range y = sin x y = cos x y = cosec x y = sec x [, ], 0, [, ] R (,), 0 R (, ) 0,

6 6 y = tan x, y = cot x 0, R R 0. Sign Convention I II III IV sin x + + cos x + + tan x + + cosec x + + sec x + + cot x + +. Behavior of Trigonometric Functions in various Quadrants I quadrant II quadrant III quadrant IV quadrant sin increases from 0 to to 0 0 to increases from to 0 cos decreases from to 0 0 to increases from to 0 increases from 0 to tan increases from 0 to increases from to 0 increase from 0 to increases from to 0 cot decrease from to 0 0 to to 0 0 to sec increases from to increase from to to to cosec decreases from to increases from to increases from to to. Basic Formulae (i) cos (x + y) = cos x cos y sin x sin y

7 7 (ii) cos (x - y) = cos x cos y + sin x sin y (iii) sin (x + y) = sin x cos y + cos x sin y (iv) sin (x y) = sin x cos y cos x sin y If none of the angles x, y and (x + y) is an odd multiple of, then (v) tan (x + y) = tanx tany tanx tany (vi) tan (x y) = tanx tany tanx tany If none of the angles x, y and (x + y) is a multiple of, then (vii) cot (x + y) = (viii) cot (x y) = cot x cot y cot x cot y cot x cot y cot y cot x. Allied Angle Relations cos x = sin x sin x = cos x con x = sin x sin x = cos x cos ( x) = cos x cos ( + x) = cos x sin ( x) = sin x sin ( + x) = sin x cos ( x) = cos x cos (n + x) = cos x sin ( x) = sin x sin (n + x) = sin x

8 8 4. Sum and Difference Formulae (i) cos x + cos y = (ii) cos x cos y = (iii) sin x + sin y = x y x y cos cos x y x y sin sin x y x y sin cos (iv) sin x sin y = x y x y cos sin (v) cos x cos y = cos (x + y) + cos (x y) (vi) sin x sin y = cos (x + y) cos (x y) (vii) sin x cos y = sin (x +y) + sin (x y) (viii) cos x sin y = sin (x + y) sin (x y) 5. Multiple Angle Formulae (i) cos x = cos x sin x = cos x = sin x = tan x tan x (ii) sin x = sin x cos x = (iii) tan x = tanx tan x (iv) sin x = sin x 4 sin³ x (v) cos x = 4 cos³ x cos x tanx tan x (vi) tan x = tanx tan x tan x 6.Trigonometric Equations No. Equations General Solution Principal value

9 9 sin θ = 0 θ = n, nz θ = 0 cos θ = 0 θ = (n + ), θ = nz tan θ = 0 θ = n θ = 0 4 sin θ = sin α θ = n + (-)ⁿ α θ = α nz 5 cos θ = cos α θ = n ± α nz θ = α, α > 0 6 tan θ = tan α θ = n + α nz θ = α 4. (i) sin θ = k = sin (n + ( )ⁿ α), n є Z θ = nπ + ( )ⁿ α, n є Z cosec θ = cosec α sin θ = sin α θ = n + ( )ⁿ α, n є Z (ii) cos θ = k = cos (n ± α), n є Z θ = n ± α, n є Z Top Diagrams. Graphs helps in visualization of properties of trigonometric functions. The graph of y = sin can be drawn by plotting a number of points (, sin ) as takes a series of different values. Since the sine function is continuous, these points can be joined with a smooth curve. Following similar procedures graph of other functions can be obtained. i. Graph of sin x

10 0 ii. Graph of cos x iii. Graph of tan x iv. Graph of sec x v. Graph of cosec x vi. Graph of cot x

This is called the horizontal displacement of also known as the phase shift.

This is called the horizontal displacement of also known as the phase shift. sin (x) GRAPHS OF TRIGONOMETRIC FUNCTIONS Definitions A function f is said to be periodic if there is a positive number p such that f(x + p) = f(x) for all values of x. The smallest positive number p for