Kerala State Board New Syllabus Plus One Maths Notes Chapter 3 Trigonometric Functions.

## Kerala Plus One Maths Notes Chapter 3 Trigonometric Functions

I. Angles

The measure of an angle is the amount of rotation performed to get the terminal side from the initial side.

1. Degree measure: If a rotation from the initial side to terminal side is \(\left(\frac{1}{360}\right)^{t h}\) of a revolution, the angle is said to have a measure of one degree, written as 1°. 1° = 60′ and f = 60″.

2. Radian measure: An angle subtended at the center by an arc of length 1 unit in a unit circle is said to be of 1 radian. Radian measure is a real number corresponding to degree measure.

180° = π radians

Radian measure = \(\frac{\pi}{180}\) × Degree measure

Degree measure = \(\frac{180}{\pi}\) × Radian measure

l = rθ, where l = arc length, r = radius of the circle and θ = angle in radian measure.

II. Trigonometric Function

Consider a unit circle with centre at the origin of the coordinate axis.

Let P (a, b) be any point on the circle which makes an angle θ° with the x-axis. Let x be the corresponding radian measure of the angle θ°, i.e; x is the arc length corresponding to θ°.

From the ∆OMP’m the figure we get;

sin θ = sin x = \(\frac{b}{1}\) = b and cos θ = cos x = \(\frac{b}{1}\) = a

This means that for each real value of x we get corresponding unique ‘sin’ and ‘cosine’ value which is also real. Hence we can define the six trigonometric functions as follows.

1. f : R → [-1, 1] defined by f(x) = sin x

2. f : R → [-1, 1] defined by f(x) = cos x

3. f : R – {nπ, n ∈ Z} → R – (-1, 1) defined by f(x) = \(\frac{1}{\sin x}\) = cosec x

4. f : R – {(2n + 1) \(\frac{\pi}{2}\)} → R – (-1, 1) defined by f(x) = \(\frac{1}{\cos x}\) = sec x

5. f : R – {(2n + 1)π, n ∈ Z} → R defined by f(x) = \(\frac{\sin x}{\cos x}\) = tan x

6. f : R – {nπ, n ∈ Z} → R defined by f(x) = \(\frac{\cos x}{\sin x}\) = cot x

Sign of trigonometric functions in different quadrants;

For odd multiple of \(\frac{\pi}{2}\) trignometric functions changes as given below.

sin → cos

cos → sin

sec → cosec

cosec → sec

tan → cot

cot → tan

The value of trigonometric functions for some specific angles;

III. Compound Angle Formula

sin(x + y) = sin x cos y + cos x sin y

sin(x – y) = sin x cos y – cos x sin y

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 – y) = sin^{2} x – sin^{2} y = cos^{2} x – cos^{2} y

cos(x + y) cos(x – y) = cos^{2} x – sin^{2} y

IV. Multiple Angle Formula

cos2x = cos^{2} x – sin^{2} x

= 1 – 2sin^{2} x

= 2 cos^{2} x – 1

= \(\frac{1-\tan ^{2} x}{1+\tan ^{2} x}\)

V. Sub-Multiple Angle Formula

VI. Sum Formula

VII. Product Formula

2 sin x cos y = sin(x + y) + sin(x – y)

2 cos x sin y = sin(x + y) – sin(x – y)

2 cos x cos y = cos(x + y) + cos(x – y)

2 sin x sin y = cos(x – y) – cos(x + y)

VIII. Solution of Trigonometric Equations

sin x = 0 gives x = nπ, where n ∈ Z

cos x = 0 gives x = (2n + 1)π, where n ∈ Z

tanx = 0 gives x = nπ, where n ∈ Z

sin x = sin y ⇒ x = nπ + (-1)^{n} y, where n ∈ Z

cos x = cos y ⇒ x = 2nπ ± y, where n ∈ Z

tan x = tan y ⇒ x = nπ + y, where n ∈ Z

Principal solution is the solution which lies in the interval 0 ≤ x ≤ 2π.

IX. Sine and Cosine formulae

Let ABC be a triangle. By angle A we mean the angle between the sides AB and AC which lies between 0° and 180°. The angles B and C are similarly defined. The sides AB, BC, and CA opposite to the vertices C, A, and B will be denoted by c, a, and b, respectively.

Theorem 1 (sine formula): In any triangle, sides are proportional to the sines of the opposite angles. That is, in a triangle ABC

\(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\)

Theorem 2 (Cosine formulae): Let A, B and C be angles of a triangle and a, b and c be lengths of sides opposite to angles A, B, and C, respectively, then

a^{2} = b^{2} + c^{2} – 2bc cos A

b^{2} = c^{2} + a^{2} – 2ca cos B

c^{2} = a^{2} + b^{2} – 2ab cos C

A convenient form of the cosine formulae, when angles are to be found are as follows: