Trigonometry For Class 11

Trigonometry is one of the major topics in Maths subject. Trigonometry deals with the measurement of angles and sides of a triangle. Usually, trigonometry is considered for the right-angled triangle. Also, its functions are used to find out the length of the arc of a circle, which forms a section in the circle with a radius and its center point.

If we break the word trigonometry, โ€˜Triโ€™ is a Greek word which means โ€˜Threeโ€™, โ€˜Gonโ€™ means โ€˜lengthโ€™, and โ€˜metryโ€™ means โ€˜measurementโ€™. So basically, trigonometry is a study of triangles, which has angles and lengths on its side. Trigonometry basics consist of sine, cosine and tangent functions. Trigonometry for class 11 contains trigonometric functions, identities to solve complex problems more simply.

Trigonometry Formulas

Here, you will learn trigonometry formulas for class 11 and trigonometric functions of Sum and Difference of two angles and trigonometric equations.

Starting with the basics of Trigonometry formulas, for a right-angled triangle ABC perpendicular at B, having an angle ฮธ, opposite to perpendicular (AB), we can define trigonometric ratios as;

Sin ฮธ = P/H

Cos ฮธ = B/H

Tan ฮธ = P/B

Cot ฮธ = B/P

Sec ฮธ = H/B

Cosec ฮธ = H/P

Where,

P = Perpendicular

B = Base

H = Hypotenuse

Trigonometry Functions

Trigonometry functions are measured in terms of radian for a circle drawn in the XY plane. Radian is nothing but the measure of an angle, just like a degree. The difference between the degree and radian is;

Degree: If rotation from the initial side to the terminal side is (1/360)th of revolution, then the angle is said to measure 1 degree.

1 degree=60minutes

1 minute=60 second

Radian: If an angle is subtended at the center by an arc of length โ€˜l, the angle is measured as radian. Supposeย ฮธ is the angle formed at the center, then

ฮธ = Length of the arc/radius of the circle.

ฮธ = l/r

Relation between Degree and Radian:

2ฯ€ radian = 360ยฐ

Or

ฯ€ radian = 180ยฐ

Where ฯ€ = 22/7

Learn more about the relation between degree and radian here.

Table for Degree and Radian relation

Degree 30ยฐ 45ยฐ 60ยฐ 90ยฐ 180ยฐ 270ยฐ 360ยฐ
Radian ฯ€/6 ฯ€/4 ฯ€/3 ฯ€/2 ฯ€ 3ฯ€/2 2ฯ€

Earlier we have discussed ofย trigonometric ratios for a degree, here we will write the table in terms of radians.

Trigonometry Table

Angle 0 ฯ€/6 ฯ€/4 ฯ€/3 ฯ€/2 ฯ€ 3ฯ€/2 2ฯ€
sin ฮธ 0 1/2 1/โˆš2 โˆš3/2 1 0 -1 0
cos ฮธ 1 โˆš3/2 1/โˆš2 1/2 0 -1 0 1
tan ฮธ 0 1/โˆš3 1 โˆš3 undefined 0 undefined 0

Trigonometry for class 11

Sign of Trigonometric Functions

sin(-ฮธ) = -sin ฮธ

cos(-ฮธ) = cos ฮธ

tan(-ฮธ) = -tan ฮธ

cot(-ฮธ) = -cot ฮธ

sec(-ฮธ) = sec ฮธ

cosec(-ฮธ) = -cosec ฮธ

Click here to know more about the sign of trigonometric functions.

Also, go through the table given below to understand the behaviour of trigonometric functions with respect to their values in different quadrants.

Quadrant I Quadrant II Quadrant III Quadrant IV
sin Increases from 0 to 1 Decreases from 1 to 0 Decreases from o to -1 Increases from -1 to 0
cos Decreases from 1 to 0 Decreases from o to -1 Increases from -1 to 0 Increases from 0 to 1
tan Increases from 0 to โˆž Increases from -โˆž to 0 Increases from 0 to โˆž Increases from -โˆž to 0
cosec Decreases from โˆž to 1 Increases from 1 to โˆž Increases from -โˆž to 1 Decreases from -1 to โˆž
sec Increases from 1 to โˆž Increases from -โˆž to 1 Decreases from -1 to โˆž Decreases from โˆž to 1
cot Decreases from โˆž to 0 Decreases from 0 to -โˆž Decreases from โˆž to 0 Decreases from 0 to -โˆž

This behaviour can be observed from the trigonometry graphs.

Read more:

Trigonometric Functions of Sum and Product of two angles

(I)

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 (ฯ€/2 – x) = cos x

cos (ฯ€/2 – x) = sin x

(II)

tan (x+y) = (tan x + tan y) /(1โˆ’tan x tan y)

tan (x-y) = (tan x โˆ’ tan y)/(1 + tan x tan y)

cot (x+y) = (cot x cot y โˆ’1)/(cot y + cot x)

cot(x-y) = (cot x cot y + 1)/( cot y โˆ’ cot x)

(III)

cos 2x = cos2 x-sin2 x = 2cos2 x-1 = 1-2sin2 x = (1-tan2 x)/(1+tan2 x)

sin 2x = 2sin x cos x= 2tan x/(1+ tan2 x)

tan 2x = 2 tan x/(1-tan2 x)

(IV)

sin 3x = 3 sin x – 4 sin3 x

cos 3x = 4 cos3ย x – 3 cos x

tan 3x = [3tan x-tan3 x]/[1-3 tan2 x]

(V)

\(\begin{array}{l}cos\ x+ cos\ y=2\ cos{(\frac{x+y}{2})}\ cos{(\frac{x-y}{2})}\\ cos\ x – cos\ y = -2\ sin{(\frac{x+y}{2})}\ sin{(\frac{x-y}{2})}\\ sin\ x + sin\ y = 2\ sin{(\frac{x+y}{2})}\ cos{(\frac{x-y}{2})}\\ sin\ x – sin\ y = 2\ cos{(\frac{x+y}{2})}\ sin{(\frac{x-y}{2})}\end{array} \)

(VI)

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

2 sin x sin y = cos (x-y) – cos (x+y)

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

2 cos x sin y = sin (x+y) – sin (x-y)

To solve the trigonometric questions for class 11, all these functions and formulas are used accordingly. By practising those questions, you can memorize the formulas as well.

Video Lessons

Basic Trigonometric Ratios & Identities

Sum, Difference & Allied Angles Formulae

Transformation of Graphs

Sum and Difference of Angles

Multiple & Sub-multiple Angles

Transformation Formulae & Conditional Identities

Solved Examples

Example 1:

Prove that sin(x+y)/ sin(xโˆ’y) = (tan x + tan y)/(tan xโ€“tan y)

Solution:

We have

LHS = sin(x+y)/sin(xโˆ’y)

= (sin x cos y + cos xย  sin y)/(sin x cos y โˆ’ cos x sin y)

Dividing numerator and denominator by cos x cos y, we get

= (tan x + tan y)/(tan xโ€“tan y) โ€”โ€”โ€”โ€“Proved.

Example 2:

Find the value of cos (31ฯ€/3).

Solution:

We know that the value of cos x repeats after the interval 2ฯ€.

Thus, cos (31ฯ€/3) = cos (10ฯ€ + ฯ€/3)

= cos ฯ€/3 = 1/2

Learn more about trigonometry by downloading BYJUโ€™S- The Learning App and get interactive videos.

Additional Trigonometry Related Articles For Class 11

Trigonometric Functions Class 11 Trigonometric Functions – Sine, Cos, Tan, Sec, Cosec & Cot
Chapter 3 – Trigonometric Functions Trigonometric Equations
Trigonometric Ratios Of Standard Angles Complementary Angles : Trigonometric Ratios
Test your Knowledge on Trigonometry For Class 11

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