Trigonometric Ratios & Identities- 1 | Mathematics (Maths) for JEE Main & Advanced PDF Download

Introduction to Trigonometry

• The word 'trigonometry' is derived from the Greek words 'trigon' and 'metron' and it means 'measuring the sides of a triangle'. The subject was originally developed to solve geometric problems involving triangles. It was studied by sea captains for navigation, surveyor to map out the new lands, by engineers and others. 
• Currently, trigonometry is used in many areas such as the science of seismology, designing electric circuits, describing the state of an atom, predicting the heights of tides in the ocean, analysing a musical tone and in many other areas.

Measurement of Angles 

There are two systems of measurement of angles:

(i) Sexagesimal or English System: Here 1 right angle = 90 (degrees)
1 = 60’ (minutes)
1’ = 60" (seconds)
(ii) Circular System: Here an angle is measured in radians. One radian corresponds to the angle subtended by an arc of length ’r ’ at the centre of the circle of radius r. It is a constant quantity and does not depend upon the radius of the circle.
(a) Relation between the two systems: Degree =  Radian*180 / π

(b) If θ is the angle subtended at the centre of a circle of radius 'r',
by an arc of length 'l' then l / r = θ

Note: l, r are in the same units and θ is always in radians.

Example.1. If the arcs of the same length in two circles subtend angles of 60° and 75° at their centres. Find the ratio of their radii.
Solution. Let r₁ and r₂ be the radii of the given circles and let their arcs of the same length s subtend angles of 60 and 75 at their centres.
Now,



⇒ 4r₁ = 5r₂ ⇒ r₁ : r₂ = 5 : 4

T-ratios (or Trigonometric Functions)

'p' is perpendicular, 'b' is base and 'h' is hypotenuse.

Note : The quantity by which the cosine falls short of unity i.e. 1 - cosθ, is called the versed sine θ of θ and also by which the sine falls short of unity i.e. 1- sinθ is called the coversed sine of θ.

➢ New Definition of T-ratios 
By using rectangular coordinates the definitions of trigonometric functions can be extended to angles of any size in the following way (see diagram). A point P is taken with coordinates (x, y). The radius vector OP has length r and the angle 0 is taken as the directed angle measured anticlockwise from the x-axis.
The three main trigonometric functions are then defined in terms of r and the coordinates x and y:

• sinθ = y / r
• cosθ = x / r
• tanθ = y / x

(The other function are reciprocals of these)
This can give negative values of the trigonometric functions.

Basic Trigonometric Identities

(i) sin θ. cosec θ = 1
(ii) cos θ. sec θ = 1
(iii) tan θ. cot θ = 1
(iv) tan θ = sin θ / cos θ & cot θ = cos θ / sin θ
(v) sin² θ + cos² θ = 1  or sin² θ = 1 - cos² θ  or cos² θ = 1 - sin² θ
(vi) sec² θ - tan² θ = 1  or sec² θ = 1 + tan² θ  or tan² θ = sec² θ  -  1
(vii) 
(viii) cosec² θ - cot² θ = 1  or  cosec² θ = 1 + cot² θ or  cot² θ  = cosec² θ - 1
(ix) 
(x) Expressing trigonometrical ratio in terms of each other:


Example.2. If sin θ + sin² θ = 1 , then prove that cos¹² θ + 3 cos¹⁰ θ + 3 cos⁸ θ + cos⁶ θ - 1 = 0
Solution. Given that sin θ = 1 - sin² θ = cos² θ

L.H.S. = cos⁶ θ (cos² θ + 1)³ - 1= sin³ θ (1 + sin θ )³ - 1= (sin θ + sin² θ)³ - 1 = 1 - 1 = 0

Example.3. 2(sin⁶ θ + cos⁶ θ) - 3 ( sin⁴ θ + cos⁴ θ) + 1 is equal to
(a) 0
(b) 1
(c) –2
(d) none of these
Ans. (a)
Solution. 2 [(sin² θ + cos² θ )³ - 3 sin² θ cos² θ ( sin² θ + cos² θ) ]  - 3 [(sin² θ + cos² θ)]² - 2sin² θ cos² θ] + 1
= 2 [1 – 3 sin² θ cos² θ] - 3 [1 - 2 sin² θ cos² θ] + 1
= 2 - 6 sin2 θ cos² θ - 3 + 6 sin² θ cos² θ + 1 = 0

Signs of Trigonometric Functions in Different Quadrants

Trigonometric Functions of Allied Angles

(a) sin (2n π + θ) = sin θ, cos (2n π + θ) = cos θ, where n ∈ I
(b)

Values of T-ratios of Some Standard Angles

N.D. → Not Defined
(a) sin nπ = 0 ; cos nπ = (-1)ⁿ; tan nπ = 0 where n ∈ I
(b) sin(2n + 1) π / 2 = (-1)ⁿ; cos(2n + 1) π / 2 = 0 where n ∈ I

Example.4. If sin θ = -1 / 2 and tan θ = 1 / √3 then θ is equal to:
(a) 30°
(b) 150°
(c) 210°
(d) None of these
Ans. (c)
Solution. Let us first find out θ lying between 0 and 360°.
Since sin θ = -1 / 2 ⇒ θ = 210° or 330° and tan θ = 1 / √3 ⇒ θ = 30° or 210°
Hence, θ = 210° or 7π / 6 is the value satisfying both.

Graph of Trigonometric Functions

(i) y = sinx

(ii) y = cosx
(iii) y = tanx
(iv) y = cotx
(v) y = secx

(vi) y = cosecx

Domains, Ranges and Periodicity of Trigonometric Functions