CBSE Class 11 Mathematics Revision Notes Chapter – 3 TRIGONOMETRIC FUNCTIONS

CBSE Class 11 Mathematics
Revision Notes
Chapter – 3
TRIGONOMETRIC FUNCTIONS

Angles
Trigonometric Functions
Sum and Difference of Two Angles
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

Quadrant:

$t-$ 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 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 $x = n\pi \pm y$

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