Trigonometric Function  Notes  Math XI
Trigonometric Function
Top Concepts
 An angle is a measure of rotation of a given ray about its initial point. The original position of the ray before rotation is called the initial side of the angle 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 rotation is anticlockwise, then the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative.
 If a rotation from the initial side to the terminal side is (1/360)^{th} of a revolution, then the angle is said to measure one degree. It is denoted by 1°.
 A degree is divided into 60 minutes, and a minute is divided into 60 seconds. One sixtieth of a degree is called a minute and is written as 1’ and one sixtieth of a minute is called a second and is written as 1”.
Thus, 1° = 60’ and 1’ = 60
 The angle subtended at the centre by an arc of length 1 unit in a unit circle is said to have a measure of 1 radian.
 Basic trigonometric ratios
Consider the following triangle:
 Some trigonometric identities
 If a point on a unit circle is on the terminal side of an angle in the standard position, then the sine of such an angle is simply the ycoordinate of the point and the cosine of the angle is the xcoordinate of that point.
 All the angles which are integral multiples of are called quadrantal angles. Values of quadrantal angles are:
cos 0 = 1, sin 0 = 0
cos = 0, sin = 1
cos π = –1, sin π = 0
cos = 0, sin = –1
cos 2π = 1, sin 2π = 0
 Even function: A function f(x) is said to be an even function if f(–x) = f(x), for all x in its domain.
 Odd function: A function f(x) is said to be an odd function if f(–x) = – f(x), for all x in its domain.
cosine is an even function and sine is an odd function
cos(–x) = cos x
sin(–x) = –sin x
 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.
In quadrant IV, only cosine function is positive.
This is depicted as follows:
 In quadrants, where the yaxis is positive (i.e. I and II), sine is positive, and in quadrants where the xaxis is positive (i.e. I and IV), cosine is positive.
 A simple rule to remember the sign of the trigonometrical ratios, in all the four quadrants, is the four letter phrase — All School To College.
 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.
 Trigonometric ratios of complementary angles
 sin (90° – θ) = cos θ
 cos (90° – θ) = sin θ
 tan (90° – θ) = cot θ
 cosec (90° – θ) = sec θ
 sec (90° – θ) = cosec θ
 cot (90° – θ) = tan θ
 Trigonometric ratios of (90° + θ) in terms of θ
 sin (90° + θ) = cos θ
 cos (90° + θ) = – sin θ
 tan (90° + θ) = – cot θ
 cosec (90° + θ) = sec θ
 sec (90° + θ) = – cosec θ
 cot (90° + θ) = – tan θ
 Trigonometric ratios of (180° – θ) in terms of θ
 sin (180° – θ) = sin θ
 cos (180° – θ) = – cos θ
 tan (180° – θ) = – tan θ
 cosec (180° – θ) = cosec θ
 sec (180° – θ) = – sec θ
 cot (180° – θ) = – cot θ
 Trigonometric ratios of (180° + θ) in terms of θ
 sin (180° + θ) = – sin θ
 cos (180° + θ) = – cos θ
 tan (180° + θ) = tan θ
 cosec (180° + θ) = – cosec θ
 sec (180° + θ) = – sec θ
 cot (180° + θ) = cot θ
 Trigonometric ratios of (360° – θ) in terms of θ
 sin (360° – θ) = – sin θ
 cos (360° – θ) = cos θ
 tan (360° – θ) = – tan θ
 cosec (360° – θ) = – cosec θ
 sec (360° – θ) = sec θ
 cot (360° – θ) = – cot θ
 Trigonometric ratios of (360° + θ) in terms of θ
 sin (360° + θ) = sin θ
 cos (360° + θ) = cos θ
 tan (360° + θ) = tan θ
 cosec (360° + θ) = cosec θ
 sec (360° + θ) = sec θ
 cot (360° + θ) = cot θ
 sin(2π + x) = sin x, so the period of sine is 2π. Period of its reciprocal is also 2π.
 cos(2π + x) = cos x, so the period of cosine is 2π. Period of its reciprocal is also 2π.
 tan(π + x) = tan x. Period of tangent and cotangent function is π.
 The graph of cos x can be obtained by shifting the sine function along the xaxis by the factor .
 The tan function differs from sine and cosine functions in two ways:
 Function tan is not defined at the odd multiples of π/2.
 The tan function is not bounded.
 Some functions and its period:
Function 
Period 
y = sin x 
2π 
y = sin (ax) 
2π/a 
y = cos x 
2π 
y = cos (ax) 
2π/a 
 For a function of the form y = kf(ax + b), the range will be ‘k’ times the range of function x, where k is any real number.
 If f(x) = sine function in above form, the range will be equal to [– k, k].
 If f(x)= cosec function in above form, the range will be equal to R – [– k, k].
 If the function is of the form ksec (ax + b) or kcosec (ax + b), the period is equal to the period of function ‘f’ divided by ‘a’.
 The position of the graph of y = kf(ax + b) is ‘b’ units to the right or left of y = f(x) depending on whether b < 0 or b > 0.
 The solutions of a trigonometric equation, for which 0 ≤ x ≤ 2π, are called principal solutions.
 The expression involving the integer ‘n’, which gives all the solutions of a trigonometric equation, is called the general solution.
 The numerically 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
 1 radian = 180°/π = 57°16' approximately.
 1° = π/180 radians = 0.01746 radians approximately.
s = rθ
Length of arc = radius × angle in radian.
This relation can only be used when θ is in radians.
 Radian measure = degree measure.
 Degree measure = radian measure.
 Values of trigonometric ratios
 Domain and range of various trigonometric functions:
 Sign convention
 Behavior of Trigonometric Functions in Various Quadrants

Quadrant I 
Quadrant II 
Quadrant III 
Quadrant IV 
sin 
increase from 0 to 1 
Decreases from 1 to 0 
Decreases from 0 to –1 
increase from –1 to 0 
cos 
Decreases from 1 to 0 
Decreases from 0 to –1 
increase from –1 to 0 
increase from 0 to 1 
tan 
increase from 0 to ∞ 
increase from –∞ to 1 
increase from 0 to ∞ 
increase from –∞ to 0 
cot 
Decreases from ∞ to 0 
Decreases from 0 to –∞ 
Decreases from ∞ to 0 
Decreases from 0 to –∞ 
sec 
increase from 1 to ∞ 
increase from –∞ to –1 
Decreases from –1 to –∞ 
Decreases from ∞ to 1 
cosec 
Decreases from ∞ to 1 
increase from 1 to ∞ 
increase from –∞ to –1 
Decreases from –1 to –∞ 
 Basic Formulae
 cos(x + y) = cos x cos y – sin x sin y
 cos(x – y) = cos x cos y + sin x sin y
 sin(x + y) = sin x cos y + cos x sin y
 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
 If none of the angles x, y and (x + y) is a multiple of π, then
 Allied Angle Relations
 cos(2π – x) = cos x
 cos(2nπ + x) = cos x
 sin(2π – x) = –sin x
 sin(2nπ + x) = sin x
 Some Important Results
 sin (x + y) sin (x – y) = sin^{2} x – sin^{2} y = cos^{2} y – cos^{2} x
 cos (x + y) cos (x + y) = cos^{2} x – sin^{2} y = cos^{2} y – sin^{2} x
 sin (x + y + z) = sin x cos y cos z + cos x sin y cos z + cos x cos y sin z + sin x sin y sin z
 cos (x + y + z) = cos x cos y cos z + sin x sin y cos z + sin x cos y sin z + cos x sin y sin z
 Sum and Difference Formulae
 2cos x cos y = cos (x + y) + cos (x – y)
 –2sin x sin y = cos (x + y) – cos (x – y)
 2sin x cos y = sin (x + y) + sin (x – y)
 2cos x sin y = sin (x + y) – sin (x – y)
 Multiple Angle Formulae:
 (i) cos 2x = cos^{2} x – sin^{2} x = 2 cos^{2} x – 1 = 1 – 2 sin^{2} x =
 sin 2x = 2 sin x cos x =
 tan 2x =
 sin 3x = 3sin x – 4sin^{3} x
 1 + cos 2x = 2cos^{2} x
 1 – cos 2x = 2sin^{2} x
 cos 3x = 4cos^{3} x – 3 cos x
 cos x ·cos 2x ·cos 2^{2}x ·cos 2^{3}x ... cos2^{n–1} x =
 Let then we have 2^{n} cos x ·cos 2x ·cos 2^{2}x ·cos 2^{3}x ... cos 2^{n–1}x = 1
 sin x ·sin (60° – x) ·sin (60° + x) =
 cos x ·cos (60° – x) ·cos (60° + x) =
 (1 + sec 2x)(1 + sec 4x)(1 + sec 8x)...(1 + sec 2^{n}x) = tan 2^{n}x cot x
 Trigonometric ratios of angle in terms of half angle
 Trigonometrical ratios of angle in terms of x/3 angle
 Trigonometrical ratios of important angles
 Trigonometric equations
 The equation acos θ + bsin θ = c is solvable for .
 Important:
 sin θ = k = sin (nα + (–1)^{n} α), n ∈ Z
θ = nπ + (–1)^{n} α, n ∈ Z
cosec θ = cosec α ⇒ sin θ = sin α
θ = nπ + (–1)^{n} α, n ∈ Z
 cos θ = k = cos (2nπ ± α), n ∈ Z
θ = 2nπ ± α, n ∈ Z
 Sine Rule: The sine rule states that
 Law of Cosine
In any △ABC,
 Projection Formulae
 a = b cos C + c cos B
 b = c cos A + a cos C
 c = a cos B + b cos A
 Napier’s Analogy (Law of Tangents)
 Area of a ABC is given by
Top diagrams
 Graphs help in the visualisation of the properties of trigonometric functions. The graph of y = sinθ can be drawn by plotting a number of points (θ, sinθ) as θ takes a series of different values. Because the sine function is continuous, these points can be joined with a smooth curve. Following similar procedures, graphs of other functions can be obtained.
 Graph of sin x
 Graph of cos x
 Graph of tan x
 Graph of sec x
 Graph of cosec x
 Graph of cot x
