# Trigonometric Functions class 11 Notes Mathematics

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## CBSE Guide Trigonometric Functions class 11 Notes

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## Trigonometric Functions class 11 Notes Mathematics

Download CBSE class 11th revision notes for Chapter 3 Trigonometric Functions class 11 Notes Mathematics in PDF format for free. Download revision notes for Trigonometric Functions class 11 Notes Mathematics and score high in exams. These are the Trigonometric Functions class 11 Notes Mathematics prepared by team of expert teachers. The revision notes help you revise the whole chapter in minutes. Revising notes in exam days is on of the best tips recommended by teachers during exam days.

CBSE Class 11 Mathematics
Revision Notes
Chapter – 3
TRIGONOMETRIC FUNCTIONS

1.   Angles
2.   Trigonometric Functions
3.   Sum and Difference of Two Angles
4.   Trigonometric Equations
• Measurement of an angle: The measure of an angle is the amount of rotation from the initial side to the terminal side.
• Right angle: If the rotating ray starting from its initial position to final position, describes one quarter of a circle, then we say that the measure of the angle formed is a right angle.
• If in a circle of radius r, an arc of length l subtends an angle of  θ radians, then l = rθ
• Radian measure = $\frac{\pi }{180}\times ~\text{Degree measure}$
• Degree measure = $\frac{180}{\pi }\times ~Radian\text{ measure}$

Trigonometric Functions

$t-$ratios                    I          II          III         IV

$\sin \theta = y$                   +         +          –           –

$\cos \theta = x$                  +         –           –           +

$\tan \theta = {y \over x}$                 +         –           +          –

• $\cos ec{\rm{ }}\theta = {1 \over {\sin \theta }}$, $\sec \theta = {1 \over {\cos \theta }}$, $\cot \theta = {1 \over {\tan \theta }}$
• Trigonometric values of some angles:
 0o 30o 45o 60o 90o 180o $\sin \theta$ 0 ${1 \over 2}$ ${1 \over {\sqrt 2 }}$ ${{\sqrt 3 } \over 2}$ 1 0 $\cos \theta$ 1 ${{\sqrt 3 } \over 2}$ ${1 \over {\sqrt 2 }}$ ${1 \over 2}$ 0 $-1$ $\tan \theta$ 0 ${1 \over {\sqrt 3 }}$ 1 $\sqrt 3$ $\infty$ 0

Trigonometric Identities:

• $\text{co}{{\text{s}}^{2}}~\text{x }+\text{ si}{{\text{n}}^{2}}\text{x }=\text{ 1}$
• $\text{1 }+\text{ ta}{{\text{n}}^{2}}\text{x }=\text{ se}{{\text{c}}^{2}}\text{x}$
• $\text{1 }+\text{ co}{{\text{t}}^{2}}\text{x }=\text{ cose}{{\text{c}}^{2}}\text{x}$

Trigonometric ratio of $\left( {{{90}^ \circ } + x } \right)$ in terms of $x$:

• $\cos (\frac{\pi }{2}+x)=\text{ -sin x}$
• $\sin (\frac{\pi }{2}+x)=\text{ cos x}$
• $\tan \left( {{\pi \over 2} + x} \right) = - \cot x$

Trigonometric ratio of $\left( {{{90}^ \circ } - x } \right)$ in terms of $x$:

• $\cos (\frac{\pi }{2}-(x)=\text{ sin x}$
• $\sin (\frac{\pi }{2}-x)=\text{ cos x}$
• $\tan \left( {{\pi \over 2} - x} \right) = \cot x$

Trigonometric ratio of $\left( {{{180}^ \circ } - x } \right)$ in terms of $x$:

• $\cos (\pi -x)=\text{ -cos x}$
• $\sin (\pi -x)=\text{ sin x}$
• $\tan \left( {\pi - x} \right) = - \tan x$

Trigonometric ratio of $\left( {{{270}^ \circ } - x } \right)$ in terms of $x$:

• $\sin \left( {{{3\pi } \over 2} - x} \right) = - \cos x$
• $\cos \left( {{{3\pi } \over 2} - x} \right) = - \sin x$
• $\tan \left( {{{3\pi } \over 2} - x} \right) = \cot x$

Trigonometric ratio of $\left( {{{270}^ \circ } + x } \right)$ in terms of $x$:

• $\sin \left( {{{3\pi } \over 2} + x} \right) = - \cos x$
• $\cos \left( {{{3\pi } \over 2} + x} \right) = \sin x$
• $\tan \left( {{{3\pi } \over 2} + x} \right) =- \cot x$

Trigonometric ratio of $\left( {{{360}^ \circ } - x } \right)$ in terms of $x$:

• $\cos \left( {2\pi - x} \right) = \cos x$
• $\sin \left( {2\pi - x} \right) = - \sin x$
• $\tan \left( {2\pi - x} \right) = - \tan x$

Trigonometric ratio of $\left( {{{360}^ \circ } + x } \right)$ in terms of $x$:

• $\cos \left( {2\pi + x} \right) = \cos x$
• $\sin \left( {2\pi + x} \right) = \sin x$
• $\tan \left( {2\pi + x} \right) = \tan x$
• $\text{cos }\left( \text{2n}\pi \text{ }+\text{ x} \right)\text{ }=\text{ cos x}$
• $~\text{sin }\left( \text{2n}\pi \text{ }+\text{ x} \right)\text{ }=\text{ sin x}$

Trigonometric Ratios of Compound Angles:

Sum Formulae:

• $\text{sin (x + y) = sin x cos y + cos x sin y}$
• $\text{cos (x + y) = cos x cos y - sin x sin y}$
• $\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}$
• $\cot (x+y)=\frac{\cot x\cot y-1}{\cot y+\cot x}$

Difference Formulae:

• $\text{sin (x - y) = sin x cos y - cos x sin y}$
• $\text{cos (x - y) = cos x cos y + sin x sin y}$
• $\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x\tan y}$
• $\cot (x-y)=\frac{\cot x\cot y+1}{\cot y-\cot x}$

Some Useful Results:

• $\sin \left( {x + y} \right)\sin \left( {x - y} \right) = {\sin ^2}x - {\sin ^2}y = {\cos ^2}y - {\cos ^2}x$
• $\cos \left( {x + y} \right)\cos \left( {x - y} \right) = {\cos ^2}x - {\sin ^2}y = {\cos ^2}y - {\sin ^2}x$
• $\tan \left( {x + y + z} \right) =$${{\tan x + \tan y + \tan z - \tan x\tan y\tan z} \over {1 - \tan x\tan y - \tan y\tan z - \tan z\tan x}}$

Transformation Formulae:

Product Formulae (on the basis of L.H.S.) or A-B formulae:

• $2\sin x\cos y = \sin \left( {x + y} \right) + \sin \left( {x - y} \right)$
• $2\cos x\sin y = \sin \left( {x + y} \right) - \sin \left( {x - y} \right)$
• $2\cos x\cos y = \cos \left( {x + y} \right) + \cos \left( {x - y} \right)$
• $2\sin x\sin y = \cos \left( {x - y} \right) - \cos \left( {x + y} \right)$

Sum and Difference Formulae (on the basis of L.H.S.) or C-D formulae:

• $\sin {\rm{C}} + \sin {\rm{D}} = 2\sin {{{\rm{C + D}}} \over 2}\cos {{{\rm{C}} - {\rm{D}}} \over 2}$
• $\sin {\rm{C}} - \sin {\rm{D}} = 2\cos {{{\rm{C + D}}} \over 2}\sin {{{\rm{C}} - {\rm{D}}} \over 2}$
• $\cos {\rm{C}} + \cos {\rm{D}} = 2\cos {{{\rm{C + D}}} \over 2}\cos {{{\rm{C}} - {\rm{D}}} \over 2}$
• $\cos {\rm{C}} - \cos {\rm{D}} = 2\sin {{{\rm{C + D}}} \over 2}\sin {{{\rm{D}} - {\rm{C}}} \over 2}$

Trigonometric Functions of Multiple and Sub-multiples of Angles:

• $\sin 2x=2\sin x\cos x=\frac{2\tan x}{1+{{\tan }^{2}}x}$
• $\cos 2x={{\cos }^{2}}x-{{\sin }^{2}}x=2{{\cos }^{2}}x-1=$ $1-2{{\sin }^{2}}x=\frac{1+{{\tan }^{2}}x}{1+{{\tan }^{2}}x}$
• $\tan 2x=\frac{2\tan x}{1-{{\tan }^{2}}x}$
• $\text{sin 3x = 3sin x - 4si}{{\text{n}}^{3}}\text{x}$
• $\text{cos 3x = 4co}{{\text{s}}^{3}}\text{x - 3cos x}$
• $\tan 3x=\frac{3\tan x-{{\tan }^{3}}x}{1-3{{\tan }^{2}}x}$
• $\sin {x \over 2} = \pm \sqrt {{{1 - \cos x} \over 2}}$
• $\cos {x \over 2} = \pm \sqrt {{{1 + \cos x} \over 2}}$
• $\tan {x \over 2} = \sqrt {{{1 - \cos x} \over {1 + \cos x}}}$
• $\sin {18^ \circ } = \cos {72^ \circ } = {{\sqrt 5 - 1} \over 4}$
• $\cos {18^ \circ } = \sin {72^ \circ } = {1 \over 4}\sqrt {10 + 2\sqrt 5 }$
• $\cos {36^ \circ } = {{\sqrt 5 + 1} \over 4}$
• $\sin {36^ \circ } = {1 \over 4}\sqrt {10 - 2\sqrt 5 }$

Trigonometric Equations:

• Principle Solutions: The solutions of a trigonometric equation, for which  are called the principle solutions.
• General Solutions: The solution, consisting of all possible solutions of a trigonometric equation is called its general solutions>
• Some General Solutions:
• $\text{sin x }=\text{ }0\text{ gives x }=\text{ n}\pi ,\text{ where n}\in \text{Z}.$
• $\text{cos x }=\text{ }0\text{ gives x }=\text{ }\left( \text{2n }+\text{ 1} \right)\text{ }\frac{\pi }{2},$$\text{ where n}\in \text{Z}.$
• $\tan x=0$ gives $x = n\pi$
• $\cot x = 0$ gives $x = \left( {2n + 1} \right){\pi \over 2}$
• $\sec x = 0$ gives no solution
• $\cos ec{\rm{ }}x$= 0 gives no solution
• $\sin x = \sin y$gives $x = n\pi + {\left( { - 1} \right)^n}y$
• $\text{cos x }=\text{ cos y},\text{ implies x }=\text{ 2n}\pi \text{ }\pm \text{ y},$$\text{ where n}\in \text{Z}.$
• $\text{tan x }=\text{ tan y implies x }=\text{ n}\pi \text{ }+\text{ y},$$\text{ where n}\in \text{Z}.$
• ${\sin ^2}x = {\sin ^2}y$ gives $x = n\pi \pm y$
• ${\cos ^2}x = {\cos ^2}y$ gives $x = n\pi \pm y$
• ${\tan ^2}x = {\tan ^2}y$ gives $x = n\pi \pm y$

## The Trigonometric Functions class 11 Notes

• CBSE Revision notes for Class 11 Mathematics PDF
• CBSE Revision notes Class 11 Mathematics – CBSE
• CBSE Revisions notes and Key Points Class 11 Mathematics
• Summary of the NCERT books all chapters in Mathematics class 11
• Short notes for CBSE class 11th Mathematics
• Key notes and chapter summary of Mathematics class 11
• Quick revision notes for CBSE exams

## CBSE Class-11 Revision Notes and Key Points

Trigonometric Functions class 11 Notes Mathematics. CBSE quick revision note for class-11 Mathematics, Physics, Chemistry, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.

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