IIT JEE Blogs

• The mathematical definition of a function from set A to set B is that to each element a ε A there exists a unique element b ε B. As we know that in direct trigonometric functions, we are given the angle and we calculate the trigonometric ratio or the value at that angle. Also for many values of the angle, the values of trigonometric ratio is same. For example for sin Θ = 1/√2 , we have

Θ =Π/4 ,5Π/4 ,  9Π/4 etc. Now, direct trigonometric functions follow the definition of a function. But in inverse trigonometry, if we say that to a certain value of the trigonometric ratio there correspond many values of the angle, it violates the definition of function as it becomes a one – many relation. That is why, some restrictions have been imposed on the angles, and these are based on the principle values of the angle.

PRE-REQUISITE

ü        If A and B are two non–empty sets then a function from A to B associated to each element x in A, a unique element f(x) in B.

f : A → B

Þ            The set A is called Domain of f.

Þ            The set B is called the co-domain of f.

Þ      The range of f is the set consisting of all the images of the element of the domain A.

Þ      If          Range of f = {f(x) : x ε A}  then the function is onto.

Þ      One-One function: If x₁, x₂ ε A then f(x₁) = f(x₂) → x₁ = x₂

Þ            Contra positively x₁ ≠ x₂ ε f(x₁) ≠ f(x₂)

Þ      Many one function: f(x₁) = f(x₂)
where x₁ ≠ x₂.

Þ      Onto function: If f(A) = B i.e.
Range = co-domain then the function is onto.

Þ      Into function: If f(A) ς B the function is into

Þ      A function is invertible iff it is a one-one onto function. The inverse of a function is defined as if y = f(x),       x ε A, y ε B and f(x) is one-one and onto in A then            x = f ^–1(y) y ε B, x ε A

Þ      If A and B are domain and range of f(x) then B and A are those
of f^–1(x).

• Trigonometric equations are natural sequel to the trigonometric ratios and identities which constitute basis of many problems in Mathematics.

• The trigonometric equations have a large number of concepts associated with relevant applications.

OBJECTIVE

• This chapter focuses on the solutions of different trigonometric equations. After studying this chapter we will learn how to find different possible solutions of a given trigonometric equation or how to find the general solution of a trigonometric equation. We will also be able to distinguish a trigonometric identity from a trigonometric equation.

PRE-REQUISITE

=>        sin (A + B) = sin A cos B + cos A sin B

=>        sin (A – B) = sin A cos B – cos A sin B

=>        cos (A + B) = cos A cos B – sin A sin B

=>        cos (A – B) = cos A cos B + sin A sin B

=>        tan (A + B) = (tanA+tanB)/ 1-tanA tanB

where A ≠ nΠ +Π/2 , B ≠ nΠ +Π/2

=>   tan (A – B) =(tanA-tanB)/1+ tanA tanB and A ± B ≠ mΠ +Π/2

=>   cot (A + B) =(cotA cot B – 1)/(cotA + cot B),

where A ≠ nΠ, B ≠n Π

=>        cot (A – B) = and A ± B≠ np

=>   sin (A + B) sin (A – B) = sin² A – sin² B

= cos² B – cos² A

=>        cos (A + B) cos (A – B) = cos² A – sin² B

= cos² B – sin² A

=>        sin2 Θ = 2 sinΘ cos Θ =2tanΘ/(1+tan 2Θ)

=>   cos 2 Θ =cos²Θ – sin² Θ =2 cos²Θ -1 = 1 – 2sin² Θ =(1-tan²Θ)/(1+tan²Θ)

=>        1 + cos 2 Θ = 2 cos²Θ,  1 – cos 2 Θ = 2 sin² Θ

or         (1+cosΘ)/2= cos²Θ, (1-cosΘ)/2 = sin² Θ

=>       tan2 Θ = 2tanΘ/(1-tan²Θ), where Θ≠ (2n + 1)Π/4

=>      (1-cosΘ)/sinΘ  = tan Θ/2, where Θ ≠ (2n + 1)Π

=>        (1+cosΘ)/ sinΘ= cot Θ/2, where Θ≠ 2n Π

=>         (1-cosΘ)/(1+cosΘ)= tan² , where Θ ≠ (2n + 1)Π

=>       (1+cosΘ)/(1-cosΘ) = cot^{2 Θ/2}, where Θ ≠ 2n Π

=>        sin 3Θ= 3sinΘ -sin³Θ

=>       cos3 Θ = 4cos³ Θ – 3cos Θ

=>           cos A cos2 A cos2² A … cos2^n-1 A =sin2″A/2″sinA

CORE CONCEPTS

• An equation involving one or more trigonometric ratios of unknown angle is called a trigonometric equation. A trigonometric equation can be written as

Q₁ (sinΘ, cosΘ, tanΘ, cotΘ, secΘ, cosecΘ)

= Q₂ (sinΘ, cosΘ, tanΘ, cotΘ, secΘ, cosecΘ)

where Q₁ and Q₂ are rational functions.

cos² x – 4 sin x = 1.

• All possible values of unknown which satisfy the given equation are called solution of the given equation.

• For complete solution “all possible values” satisfying the equation must be obtained.

• This is trigonometric equation as it is not satisfied for all values of x e.g.,  does not satisfy the given equation.

Identify whether the following are trigonometric equations or trigonometric identities.

1. sin³ A = 3sin A – 4 sin³ A

Sol. Trigonometric Identity

2. cos⁷ x + sin⁴ x = 1

Sol. Trigonometric Equation

• Constant function is a periodic function without any period. This happens because of the non-existence of the least positive real number which is due to the continuity of real number system.

• If f(x) has it’s period T then f(ax + b) has its period .

• If f(x) has its period T₁ and g(x) has its period                T₂ then   (af(x) + bg(x))  has  its  period                         £ L.C. M.(T₁, T₂).  Moreover if f(x) and g(x) are basic trigonometric functions then period of                        [af(x) + bg(x)]  =  L.C.M. (T₁, T₂)

Examine whether sin x is a periodic function or not. If so, find its period.

Given f(x) = sin x. Let’s assume sin x to be periodic. So, it must have some positive value independent of x say T such that f(x + T) = f(x)

• sin (x + T) = sin x

• x + T = n p + (–1)ⁿ x where n = 0, ± 1, ± 2 ……

The positive values of T independent of x are given by n p where n = 2, 4, 6……..

Further according to definition for periodic number, it should be least. So, here we have T = 2p.

Thus, it is proved that sin x is periodic function having periodicity 2p.

Prove that f(x) = sin√x  is not a periodic function.

Proof : Let the positive real number T be such that f (x + T) = f(x)

This above relation does not give any positive value of T independent of x because it holds only when T = 0.

f(x) is non-periodic function.

Let f(x) = x – [x] where [x] is the greatest integer less than or equal to x. Find out the periodicity of f(x). Assume f(x + T) = f(x)

• (x + T) – [x + T] = x- [x]

• T = [x + T] – [x] = an integer.

Hence least positive value of T independent of x is 1.

Thus f(x) is a periodic function of period 1.

This can be explained through graphs. As in the case of algebraic function, we can have same idea about the nature of a trigonometric function by its graph.

The variations in the values of the trigonometric ratios generated the concept of graph in trigonometric functions.

From the graph, we observe that:

• The value of sin x repeats itself after an interval of 2p. So sin x is a periodic function with period of 2p. Actually a revolution of 2p is the complete revolution.

• sin x takes value from –1 to 1.

INTRODUCTION

Trigonometry is the subject which deals with the properties of triangles. You might recall from your previous knowledge that in a right angled triangle the ratios between any two sides may be defined in terms of what are called trigonometric ratios, but the scope of trigonometry is not limited to this only. These rules or laws of trigonometry are used as tools for mathematical analysis of various problems in physics and engineering.

Trigonometric functions

A real number Θ can be interpreted as the measure of the angle constructed as follows: wrap a piece of string of length Θ units around the unit circle  x² + y² = 1 (counterclockwise if Θ≥ 0, clockwise if Θ < 0) with initial point P(1, 0) and terminal point Q(x, y). This gives rise to the central angle with vertex O(0, 0) and sides through the points P and Q. All six trigonometric functions of Θ are defined in terms of the coordinates of the point Q(x,y), as follows:

cos Θ = x                                                       sec Θ = 1/x if x ≠ 0

sin Θ = y                                                       cosec Θ = 1/y, if y ≠ 0

tan Θ = y/x if x ≠ 0                                         cot Θ = x/y if y ≠ 0

Since Q(x,y) is a point on the unit circle, we know that x² + y² = 1. This fact and the definitions of the trigonometric functions give rise to the following fundamental identities:

Pythagorean identity           sin² Θ + cos² Θ = 1

Reciprocal identities

sec Θ = 1/cos Θ

cosec Θ = 1/sin Θ

tan Θ = sin Θ/cos Θ

tan Θ = 1/cot Θ

cot Θ = 1/tan Θ

This modern notation for trigonometric functions is due to L. Euler (1748).

More generally, if Q(x,y) is the point where the circle x² + y² = R² of radius R is intersected by the angle Θ, then it follows (from similar triangles) that

cos Θ = x/R                                             sec Θ = R/x, if x ≠ 0

sin Θ= y/R                                               cosec Θ = R/y, if y ≠ 0

tan Θ = y/x, if x ≠ 0                                 cot Θ = y/x if y ≠ 0

Periodic Functions

If an angle Θ corresponds to a point Q(x, y) on the unit circle, it is not hard to see that the angle Θ + 2Π corresponds to the same point Q(x, y), and hence that

cos (Θ + 2Π) = cos Θ,  sin (Θ + 2Π) = sin Θ …(1)

Moreover, 2Π is the smallest positive angle for which Equations (1) are true for any angle Θ. In general, we have for all angles Θ:

cos (Θ + 2nΠ) = cos Θ, sin (Θ + 2nΠ) = sin Θ, n = 0, ±1, ±2 ….                                                                                       …(2)

We call the number 2Πthe period of the trigonometric functions sin and cos, and refer to these functions as being periodic. Both sec and cosec are periodic functions as well, with period 2Π, while tan and cot are periodic with period Π.

Illustrations

1. Find the period of the function
f(x) = – 3 cos (3x).

Sol. The function f(x) = – 3 cos (3x) runs through a full cycle when the angle 3x runs from 0 to 2Π, or equivalently when x goes from 0 to 2Π/3. The period of f(x) is then 2Π/3.

2. Find the period of the function f(t) = 8 sin (7Πt).

Sol. For the function f(t) = 8 sin (7Πt) to run through a full cycle, the angle 7Πt should run from 7Πt = 0 to 7Πt = 2Π, and hence t should run from t = 0 to t = 2/7. The period of f(t) is then 2/7.

Algebra:

Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations.
Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots

Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers.
Logarithms and their properties.
Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients.
Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.
Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations.

Trigonometry:

Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles, general solution of trigonometric equations.
Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle, inverse trigonometric functions (principal value only).

Analytical geometry:

Two dimensions: Cartesian coordinates, distance between two points, section formulae, shift of origin.
Equation of a straight line in various forms, angle between two lines, distance of a point from a line; Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines; Centroid, orthocentre, incentre and circumcentre of a triangle. Equation of a circle in various forms, equations of tangent, normal and chord.
Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal. Locus Problems.
Three dimensions: Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.

Differential calculus: Real valued functions of a real variable, into, onto and one-to-one functions, sum, difference, product and quotient of two functions, composite functions, absolute value, polynomial, rational, trigonometric, exponential and logarithmic functions.
Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, L’Hospital rule of evaluation of limits of functions.
Even and odd functions, inverse of a function, continuity of composite functions, intermediate value property of continuous functions. Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions.
Derivatives of implicit functions, derivatives up to order two, geometrical interpretation of the derivative, tangents and normals, increasing and decreasing functions, maximum and minimum values of a function, Rolle’s Theorem and Lagrange’s Mean Value Theorem.

Integral calculus: Integration as the inverse process of differentiation, indefinite integrals of standard functions, definite integrals and their properties, Fundamental Theorem of Integral Calculus. Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas involving simple curves. Formation of ordinary differential equations, solution of homogeneous differential equations, separation of variables method, linear first order differential equations.

Vectors: Addition of vectors, scalar multiplication, dot and cross products, scalar triple products and their geometrical interpretations.