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```Basic Concepts of Trigonometric Functions

Trigonometric functions or the circular functions are basically the functions of an angle. They establish the relationship between the sides and angles of the triangle. The most elementary trigonometric functions are sine, cosine and tangent. All the remaining ratios can be derived through these three only.

These ratios have got some implications geometrically. In a unit circle, if we form a triangle with a ray from the origin and making some angle with the x- axis, then the sine ratio will depict the y- component, while the cosine denotes the x- component. Since the tangent i.e. tan is the ratio of y by x, so the slope is given by the tan. More explicit definitions suitable for applications are given below:

These are the basic formulae of the trigonometric functions. The functions in the second column are simply the reciprocal of those in the first. These formulae are vital in solving some direct IIT JEE trigonometry questions.

Another important point to be noted is the sign of these trigonometric functions. The sign varies depending on the quadrant in which they fall.

The figure below illustrates the signs of the trigonometric functions in various quadrants. The knowledge of the signs is vital in solving the problems.

For more comprehensive explanation on signs of the functions, refer the video

Concept of Periodicity of Trigonometric functions

Periodicity, as the name suggests is the property of a function to repeat itself in a specific format at regular intervals. All trigonometric functions are periodic. The literal definition of a periodic function is as follows:

A function f(x) is said to be periodic if there exists some p such that f(x) = f(x + p) for all x and some fixed p. Here p is called the period of f. This concept is very useful as once the period of a function is known, it becomes possible to know the value of a function at any point. We list below the major functions with their periods:

Sin and cos both have 2Π as their period.

Tan function has Π as the period.

The reciprocal functions have the same period as that of the original functions.

If the period of f(x) is T then that of kf(ax+b) is T/mod (a), hence period is affected by coefficient of x only.

If f(x) has its period T and g(x) has its period Mthen (a f(x) + bg(x) has its period < L.C.M. (T, M). Moreover if f(x) and g(x) are basic trigonometric functions then period of [af (x) + bg(x)] = L.C.M. (T, M).

If a constant is added, subtracted, multiplied or divided in a periodic function, its period is unaffected.

Remark:

Before going to the graphical representation, another concept that would be needed is Amplitude.

Amplitude of a periodic function is the maximum absolute value of the vertical component of the function.

For More Clarifications please refer to the following video

Graphical Representation of trigonometric functions

The concept of periodicity lays the groundwork for graphical representation of trigonometric functions. Through graphs, the picture of the function becomes quite clear and then questions can be cracked easily. Some of the graphs of trigonometric functions are illustrated below:

We start with the basic function, i.e. sin t. Now we know its period is 2Π so it will repeat after every 2Π units and its amplitude is 1. Plotting the graph, considering all these points,

On the same lines, by just altering the amplitude and the period, the graph of sin nt function can be drawn, where n= 1,2,3,... The shape will always remain the same.

Fig. 1 Here f(x)= sin x and g(x)= sin 2x

Fig. 2 Here f(x) = sin x and g(x) = 2sin x

We plot f(x) = cos x. Again, since the period is 2Π, so the graph will repeat after 2Π units.

f(x)= tan x

f(x)= cot x

f(x)= sec x

For more conceptual clarity on periodicity and graphical representation please refer the video

Illustration 1:

Find all trigonometric functions of an angle  in the third quadrant for which
Solution: Before proceeding towards the solution, draw the figure so as to have a clear picture of the question.

As in the Figure, cosθ= x/R= -5/6 and also we may consider x= -5 and R= 6.
Since, x2+y2=62=R2=36 we find that y= -√R2-x2= -√11

Note that the negative signs indicate the third quadrant. And hence all the rest ratios follow

Illustration 2: Evaluate the value of the expression

3[sin4(3π/2 - α) + sin4(3π + α)] – 2 [sin6(π/2 + α) + sin6(5π - α)].

Solution: The given expression is

3[sin4(3π/2 - α) + sin4(3π + α)] – 2 [sin6(π/2 + α) + sin6(5π - α)].

= 3[(cos4α + sin4 α) – 2(cos6α + sin6 α)]

= 3 [(1-2sin2α cos2α) -2(1-3sin2α cos2α)] = 1.

Illustration 3: Find the value of

3(sin x – cos x)4 + 6(sin x + cos x)2 + 4(sin6x + cos6x).

Solution: The given expression is

3(sin x – cos x)4 + 6(sin x + cos x)2 + 4(sin6x + cos6x)

= 3 (1 - sin 2x)2 + 6(1 + sin 2x) + 4[(sin2x + cos2x)3 -3sin2x cos2x (sin2x + cos2x)]

= 3 (1 - 2sin2x + sin22x) + (6 + 6 sin2x) + 4(1-3sin2x cos2x)

= 3 (1- 2sin2x + sin22x + 2 + 2sin2x) + 4 [1- ¾ sin22x]

= 13 + 3 sin22x – 3 sin22x = 13.

Illustration 4: If cos (a + b) = 4/5, sin (a-b) = 5/13 and a and b lie between 0 to π/4, find tan 2a.

Solution: It is given that cos (a + b) = 4/5, sin (a-b) = 5/13

It follows that sin (a + b) = 3/5, cos (a-b) = 12/13

Hence, tan (a+b) = 3/4 and tan (a-b) = 5/12

Hence, this implies tan [(a+b) + a-b] =

[tan (a+b) + tan (a-b)]/ [1 + tan(a+b)tan(a-b)]

= (3/4 + 5/12)/ (1 – 3/4. 5/12)

Hence, tan 2a = 14/12. 48/33 = 56/33.