Karnataka Class 11 Commerce Maths Trigonometric Functions

Karnataka Class 11 Commerce Maths Trigonometric Functions

Karnataka Class 11 Commerce Maths Trigonometric Functions : Trigonometry is a branch of Mathematics. The word trigonometry is derived from Greek words tri-gono & metron (tri-three, gono=angle, metron=measurement.) Thus the word trigonometry, literally means measurement of triangles. Primarily the subject deals with the relations between the sides, angles and area of a triangle. But trigonometry has a much wider scope and is used in geometrical and algebraic investigation in pure and applied mathematics.

Hipparchus, a Greek astronomer, is known as a founder of trigonometry and others, who contributed to this branch include Euler, John-Bernolli, Fourier, and Gauss. The Foremost Indian Mathematicians who have worked on this branch of mathematics are Aryabhatta, Brahmagupta and Bhaskaracharya.

Trigonometry is a branch of mathematics which studies the relationships that involve angles and lengths of triangles. It is very important to mathematics subject as an element of statistics, linear algebra and calculus. In addition to mathematics, it also contributes majorly to engineering, physics, astronomy and architectural design. Trigonometry Formulas for class 11 plays crucial role in solving any problem related to this chapter.

Karnataka Class 11 Commerce Maths Trigonometric Functions

Trigonometric Functions
Angle: Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another.

Definition of trigonometric functions with the help of unit circle.

Truth of the identity
sin2x + cos2
x = 1, for all x.

Signs of trigonometric functions and sketch of their graphs.

Trigonometric functions of sum and difference of two angles:

Deducing the formula for cos(x+y) using unit circle .

Expressing sin ( x+ y ) and cos ( x + y ) in terms of sin x, sin y, cos x and cos y .

Definition of allied angles and obtaining their trigonometric ratios using compound angle formulae.

Trigonometric ratios of multiple angles:

Identities related to sin2x, cos2x, tan2x, sin3x, cos3x and tan3x.

Karnataka Class 11 Commerce Maths Trigonometric Functions

Trigonometric Equations:

General solution of trigonometric equations of the type sinθ = sin α, cosθ = cosα and tanθ = tan α. and problems.

Proofs and simple applications of sine and cosine rule.

Angle: When a line revolves about a point O from one position P to another position Q. We say an angle is formed and this angle is denoted by /POQ. The point O is called the vertex of the angle.

A radian is an angle subtended at the centre of the circle, by an arc, whose length is equal to radius of the circle.

From the figure OA = O B = r =arc AB then/ AOB = 1C (One Radian is denoted by 1C)

Now, we shall show that a radian, constructed according to above definition is of constant magnitude.

Karnataka Class 11 Commerce Maths Trigonometric Functions

Trigonometic Ratios:

Let ∠ BOC = θ, be any acute angle, B be any point on OC, one of the boundary lines and BC is perpendicular to OC. Thus we have a right angled triangle BOC.

With respect to the angle θ, the side BC is called the opposite side, OC is called adjacent side and OB is the hypotenuse of the triangle.

With respect to the angle θ, we define the following six ratios called trignomotric ratios.

Let us now define the 6 trigonometric ratios as follows.

Karnataka Class 11 Commerce Maths Trigonometric Functions

1. Sine of θ written as sin θ = BC / OB = Opposite side / Hypotenuse side

2. Cosine of  θ  written as cos  θ  = OC / OB =  Adjacent side / Hypotenuse side

3. tangent of θ written as tan θ =  BC / OC = Opposite side / Adjacent side

4. Cosecant of θ written as cosec θ = OB / BC = Hypotenuse side / Opposite side

5. Secant of θ written as sec θ  = OB / OC = Hypotenuse side / Adjacent side

6. Cotangent of θ written as cot θ = OC / BC =  Adjacent side / Opposite side

The above definitions of trigonometric ratios remains unaltered as long as the angle remains the same, i.e. trigonometric ratios depends on the angle but not on the lengths of the sides of the right angled triangle.

Karnataka Class 11 Commerce Maths Trigonometric Functions

Note: We know that in every right angled triangle the hypotenuse is the greatest side, thus it follows from the definition of trigonometric ratios that those ratios which have the hypotenuse in the denominator can never be greater than unity, while those which have hypotenuse in the numerator can never be less than unity.

Further those ratios which do not involve the hypotenuse may have any numerical value. Thus we have the following results.
 The sin θ and cos θ of an angle can never be greater than 1.
 The cosec θ and sec θ of an angle can never be less than 1.
 The tan θ and cot θ of an angle may have any numerical value.

Karnataka Class 11 Commerce Maths Trigonometric Functions

Relation between the trigonometric ratios.

In the right angled triangle as shown, consider:
Hypotenuse = r
Adjacent side = x
Opposite side = y

1. Prove that sin θ . cosec θ = 1
From a right angled triangle shown below we have:
Sin θ = y / r 
Cosec θ = r / y
L.H.S = sin θ . cosec θ= y / r . r / y = 1
Therefore, sin θ . cosec θ = 1
⇒ cosec θ = 1 / sinθ or sin θ = 1 / cosec θ

Similarly we can prove the following results:

2. cos θ . sec θ = 1

3. tan θ . cot θ = 1

4. Prove that tan θ = sin θ / cos θ
> R.H.S = sin θ / cos θ = y/r ÷ x/r = y/x = tan θ = L.H.S
Therefore tan θ = sin θ / cos θ.

Karnataka Class 11 Commerce Maths Trigonometric Functions

Basic Formulas

Karnataka Class 11 Commerce Maths Trigonometric Functions

Signs of Trigonometric ratios:

The trigonometric functions of any angle are already defined as sin θ = y/r  

cos θ = x/r  and 

tan θ = y/x  and the others are reciprocals of these, since r is always positive.

The algebraic sign of these numbers depend on x & y only. Now let us consider the different positions of the terminal ray of angle .

Karnataka Class 11 Commerce Maths Trigonometric Functions

We have listed all the important trigonometric identities here so that students don’t miss out any:

· Pythagorean Identities

sin ²X + cos ²X = 1

1 + tan ²X = sec ²X

1 + cot ²X = csc ²X

·  Negative Angle Identities

sin (-X) = – sin X, odd function

csc (-X) = – csc X, odd function

cos (-X) = cos X, even function

sec (-X) = sec X, even function

tan (-X) = – tan X, odd function

cot (-X) = – cot X, odd function

· Cofunctions Identities

sin (π /2 – X) = cos X

cos (π /2 – X) = sin X

tan (π /2 – X) = cot X

cot (π/2 – X) = tan X

sec (π /2 – X) = csc X

csc (π /2 – X) = sec X

· Addition Formulas

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 cosY – cos X sin Y

tan (X + Y) = [ tan X + tan Y ] / [ 1 – tan X tan Y]

tan (X – Y) = [ tan X – tan Y ] / [ 1 + tan X tan Y]

cot (X + Y) = [ cot X cot Y – 1 ] / [ cot X + cot Y]

cot (X – Y) = [ cot X cot Y + 1 ] / [ cot Y – cot X]

· Sum to Product Formulas

cos X + cos Y = 2cos[(X + Y)/ 2] cos[(X – Y)/ 2]

sin X + sin Y = 2sin[(X + Y)/ 2] cos[(X – Y)/ 2]

· Difference to Product Formulas

cos X – cos Y = – 2sin[(X + Y) / 2] sin[(X – Y) / 2]

sin X – sin Y = 2cos[(X + Y) / 2] sin[(X – Y) / 2]

· Product to Sum/Difference Formulas

cos X cos Y = (1/2) [cos (X – Y) + cos (X + Y)]

sin X cos Y = (1/2) [sin (X + Y) + sin (X – Y)]

cos X sin Y = (1/2) [sin (X + Y) – sin[ (X – Y)]

sin X sin Y = (1/2) [cos (X – Y) – cos (X + Y)]

· Difference of Squares Formulas

sin ²X – sin ²Y = sin (X + Y) sin (X – Y)

cos ²X – cos ²Y = – sin (X + Y) sin (X – Y)

cos ²X – sin ²Y = cos (X + Y) cos (X – Y)

· Double Angle Formulas

sin (2X) = 2 sin X cos X

cos (2X) = 1 – 2sin ²X = 2cos ²X – 1

tan (2X) = 2tan X/[1 – tan ²X]

·  Multiple Angle Formulas

sin (3X) = 3sin X – 4sin ³X

cos (3X) = 4cos ³X – 3cos X

sin (4X) = 4sin X cos X – 8sin ³X cos X

cos (4X) = 8cos ⁴X – 8cos ²X + 1

· Half Angle Formulas

sin (X/2) = ±√[(1 – cos X)/2]

cos (X/2) = ±√[(1 + cos X)/2]

tan (X/2) = ±√[(1 – cos X)/(1 + cos X)]

= sin X/(1 + cos X)

= (1 – cos X)/sin X

· Power Reducing Formulas

sin ²X = 1/2 – (1/2) cos (2X))

cos ²X = 1/2 + (1/2) cos (2X))

sin ³X = (3/4) sin X – (1/4) sin (3X)

cos ³X = (3/4) cos X + (1/4) cos (3X)

sin ⁴X = (3/8) – (1/2)cos (2X) + (1/8)cos (4X)

cos ⁴X = (3/8) + (1/2)cos (2X) + (1/8)cos (4X)

sin ⁵X = (5/8)sin X – (5/16)sin (3X) + (1/16)sin (5X)

cos ⁵X = (5/8)cos X + (5/16)cos (3X) + (1/16)cos (5X)

sin ⁶X = 5/16 – (15/32)cos (2X) + (6/32)cos (4X) – (1/32)cos (6X)

cos ⁶X = 5/16 + (15/32)cos (2X) + (6/32)cos (4X) + (1/32)cos (6X)

· Trigonometric Functions Periodicity

sin (X + 2π) = sin X, period 2π

cos (X + 2π) = cos X, period 2π

sec (X + 2π) = sec X, period 2π

cosec (X + 2π) = cosec X, period 2π

tan (X + π) = tan X, period π

cot (X + π) = cot X, period π

Trigonometric functions and trigonometry ratios are some of the most imperative areas of trigonometry. Trigonometry is the branch of Mathematics that deals with triangles and the relationships between the sides and angles. The trigonometry functions are universal in parts of pure mathematics and applied mathematics which also lay the groundwork for many branches of science and technology.

To download all the important trigonometric identities check here 

Trigonometric functions and trigonometry ratios

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