Trigonometric Functions - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes
Chapter - 3
TRIGONOMETRIC FUNCTIONS

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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 = π180× Degree measure$\frac{\pi }{180}\times \text{Degree measure}$
• Degree measure = 180π× Radian measure$\frac{180}{\pi }\times Radian\text{ measure}$

Trigonometric Functions

• Quadrant:

          t−$t-$ratios                    I          II          III         IV

          sinθ=y$\sin \theta =y$                   +         +          -           -

          cosθ=x$\cos \theta =x$                  +         -           -           +

          tanθ=yx$\tan \theta =\frac{y}{x}$                 +         -           +          -

• cosecθ=1sinθ$\cos ec\theta =\frac{1}{\sin \theta }$, secθ=1cosθ$\sec \theta =\frac{1}{\cos \theta }$, cotθ=1tanθ$\cot \theta =\frac{1}{\tan \theta }$
• Trigonometric values of some angles:

	0o	30o	45o	60o	90o	180o
sinθ$\sin \theta$	0	12$\frac{1}{2}$	12√$\frac{1}{\sqrt{2}}$	3√2$\frac{\sqrt{3}}{2}$	1	0
cosθ$\cos \theta$	1	3√2$\frac{\sqrt{3}}{2}$	12√$\frac{1}{\sqrt{2}}$	12$\frac{1}{2}$	0	−1$-1$
tanθ$\tan \theta$	0	13√$\frac{1}{\sqrt{3}}$	1	3–√$\sqrt{3}$	∞$\mathrm{\infty }$	0

Trigonometric Identities:

• cos2 x + sin2x = 1$\text{co}{\text{s}}^2\text{x }+\text{ si}{\text{n}}^2\text{x }=\text{ 1}$

• 1 + tan2x = sec2x$\text{1 }+\text{ ta}{\text{n}}^2\text{x }=\text{ se}{\text{c}}^2\text{x}$
• 1 + cot2x = cosec2x$\text{1 }+\text{ co}{\text{t}}^2\text{x }=\text{ cose}{\text{c}}^2\text{x}$

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

• cos(π2+x)= -sin x$\cos (\frac{\pi }{2}+x)=\text{ -sin x}$
• sin(π2+x)= cos x$\sin (\frac{\pi }{2}+x)=\text{ cos x}$
• tan(π2+x)=−cotx$\tan \left(\frac{\pi }{2}+x\right)=-\cot x$

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

• cos(π2−(x)= sin x$\cos (\frac{\pi }{2}-(x)=\text{ sin x}$
• sin(π2−x)= cos x$\sin (\frac{\pi }{2}-x)=\text{ cos x}$
• tan(π2−x)=cotx$\tan \left(\frac{\pi }{2}-x\right)=\cot x$

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

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

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

• sin(3π2−x)=−cosx$\sin \left(\frac{3\pi }{2}-x\right)=-\cos x$
• cos(3π2−x)=−sinx$\cos \left(\frac{3\pi }{2}-x\right)=-\sin x$
• tan(3π2−x)=cotx$\tan \left(\frac{3\pi }{2}-x\right)=\cot x$

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

• sin(3π2+x)=−cosx$\sin \left(\frac{3\pi }{2}+x\right)=-\cos x$
• cos(3π2+x)=sinx$\cos \left(\frac{3\pi }{2}+x\right)=\sin x$
• tan(3π2+x)=−cotx$\tan \left(\frac{3\pi }{2}+x\right)=-\cot x$

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

• cos(2π−x)=cosx$\cos \left(2\pi -x\right)=\cos x$
• sin(2π−x)=−sinx$\sin \left(2\pi -x\right)=-\sin x$
• tan(2π−x)=−tanx$\tan \left(2\pi -x\right)=-\tan x$

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

• cos(2π+x)=cosx$\cos \left(2\pi +x\right)=\cos x$
• sin(2π+x)=sinx$\sin \left(2\pi +x\right)=\sin x$
• tan(2π+x)=tanx$\tan \left(2\pi +x\right)=\tan x$
• cos (2nπ + x) = cos x$\text{cos }\left(\text{2n}\pi +\text{ x}\right)=\text{ cos x}$
•  sin (2nπ + x) = sin x$\text{sin }\left(\text{2n}\pi +\text{ x}\right)=\text{ sin x}$

Trigonometric Ratios of Compound Angles:

Sum Formulae:

• sin (x + y) = sin x cos y + cos x sin y$\text{sin (x + y) = sin x cos y + cos x sin y}$
• cos (x + y) = cos x cos y - sin x sin y$\text{cos (x + y) = cos x cos y - sin x sin y}$
• tan(x+y)=tanx+tany1−tanxtany$\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y}$
• cot(x+y)=cotxcoty−1coty+cotx$\cot (x+y)=\frac{\cot x\cot y-1}{\cot y+\cot x}$

Difference Formulae:

• sin (x - y) = sin x cos y - cos x sin y$\text{sin (x - y) = sin x cos y - cos x sin y}$
• cos (x - y) = cos x cos y + sin x sin y$\text{cos (x - y) = cos x cos y + sin x sin y}$
• tan(x−y)=tanx−tany1+tanxtany$\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x\tan y}$
• cot(x−y)=cotxcoty+1coty−cotx$\cot (x-y)=\frac{\cot x\cot y+1}{\cot y-\cot x}$

Some Useful Results:

• sin(x+y)sin(x−y)=sin2x−sin2y=cos2y−cos2x$\sin \left(x+y\right)\sin \left(x-y\right)={\sin }^{2}x-{\sin }^{2}y={\cos }^{2}y-{\cos }^{2}x$
• cos(x+y)cos(x−y)=cos2x−sin2y=cos2y−sin2x$\cos \left(x+y\right)\cos \left(x-y\right)={\cos }^{2}x-{\sin }^{2}y={\cos }^{2}y-{\sin }^{2}x$
• tan(x+y+z)=$\tan \left(x+y+z\right)=$tanx+tany+tanz−tanxtanytanz1−tanxtany−tanytanz−tanztanx$\frac{\tan x+\tan y+\tan z-\tan x\tan y\tan z}{1-\tan x\tan y-\tan y\tan z-\tan z\tan x}$

Transformation Formulae:

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

• 2sinxcosy=sin(x+y)+sin(x−y)$2\sin x\cos y=\sin \left(x+y\right)+\sin \left(x-y\right)$
• 2cosxsiny=sin(x+y)−sin(x−y)$2\cos x\sin y=\sin \left(x+y\right)-\sin \left(x-y\right)$
• 2cosxcosy=cos(x+y)+cos(x−y)$2\cos x\cos y=\cos \left(x+y\right)+\cos \left(x-y\right)$
• 2sinxsiny=cos(x−y)−cos(x+y)$2\sin x\sin y=\cos \left(x-y\right)-\cos \left(x+y\right)$

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

• sinC+sinD=2sinC+D2cosC−D2$\sin \mathrm{C}+\sin \mathrm{D}=2\sin \frac{\mathrm{C}+\mathrm{D}}{2}\cos \frac{\mathrm{C}-\mathrm{D}}{2}$
• sinC−sinD=2cosC+D2sinC−D2$\sin \mathrm{C}-\sin \mathrm{D}=2\cos \frac{\mathrm{C}+\mathrm{D}}{2}\sin \frac{\mathrm{C}-\mathrm{D}}{2}$
• cosC+cosD=2cosC+D2cosC−D2$\cos \mathrm{C}+\cos \mathrm{D}=2\cos \frac{\mathrm{C}+\mathrm{D}}{2}\cos \frac{\mathrm{C}-\mathrm{D}}{2}$
• cosC−cosD=2sinC+D2sinD−C2$\cos \mathrm{C}-\cos \mathrm{D}=2\sin \frac{\mathrm{C}+\mathrm{D}}{2}\sin \frac{\mathrm{D}-\mathrm{C}}{2}$

Trigonometric Functions of Multiple and Sub-multiples of Angles:

• sin2x=2sinxcosx=2tanx1+tan2x$\sin 2x=2\sin x\cos x=\frac{2\tan x}{1+{\tan }^{2}x}$
• cos2x=cos2x−sin2x=2cos2x−1=$\cos 2x={\cos }^{2}x-{\sin }^{2}x=2{\cos }^{2}x-1=$ 1−2sin2x=1+tan2x1+tan2x$1-2{\sin }^{2}x=\frac{1+{\tan }^{2}x}{1+{\tan }^{2}x}$
• tan2x=2tanx1−tan2x$\tan 2x=\frac{2\tan x}{1-{\tan }^{2}x}$
• sin 3x = 3sin x  - 4sin3x$\text{sin 3x = 3sin x - 4si}{\text{n}}^3\text{x}$
• cos 3x = 4cos3x - 3cos x$\text{cos 3x = 4co}{\text{s}}^3\text{x - 3cos x}$
• tan3x=3tanx−tan3x1−3tan2x$\tan 3x=\frac{3\tan x-{\tan }^{3}x}{1-3{\tan }^{2}x}$
• sinx2=±1−cosx2−−−−−√$\sin \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{2}}$
• cosx2=±1+cosx2−−−−−√$\cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}$
• tanx2=1−cosx1+cosx−−−−−√$\tan \frac{x}{2}=\sqrt{\frac{1-\cos x}{1+\cos x}}$
• sin18∘=cos72∘=5√−14$\sin {18}^{\circ }=\cos {72}^{\circ }=\frac{\sqrt{5}-1}{4}$
• cos18∘=sin72∘=1410+25–√−−−−−−−−√$\cos {18}^{\circ }=\sin {72}^{\circ }=\frac{1}{4}\sqrt{10+2\sqrt{5}}$
• cos36∘=5√+14$\cos {36}^{\circ }=\frac{\sqrt{5}+1}{4}$
• sin36∘=1410−25–√−−−−−−−−√$\sin {36}^{\circ }=\frac{1}{4}\sqrt{10-2\sqrt{5}}$

Trigonometric Equations:

• Principle Solutions: The solutions of a trigonometric equation, for which 0≤x<2π$0\le x<2\pi$ 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:
• sin x = 0 gives x = nπ, where n∈Z.$\text{sin x }=0\text{ gives x }=\text{ n}\pi ,\text{ where n}\in \text{Z}.$
• cos x = 0 gives x = (2n + 1) π2,$\text{cos x }=0\text{ gives x }=\left(\text{2n }+\text{ 1}\right)\frac{\pi }{2},$ where n∈Z.$\text{ where n}\in \text{Z}.$
• tanx=0$\tan x=0$ gives x=nπ$x=n\pi$
• cotx=0$\cot x=0$ gives x=(2n+1)π2$x=\left(2n+1\right)\frac{\pi }{2}$
• secx=0$\sec x=0$ gives no solution
• cosecx$\cos ecx$= 0 gives no solution
• sinx=siny$\sin x=\sin y$gives x=nπ+(−1)ny$x=n\pi +{\left(-1\right)}^{n}y$
• cos x = cos y, implies x = 2nπ ± y,$\text{cos x }=\text{ cos y},\text{ implies x }=\text{ 2n}\pi \pm \text{ y},$ where n∈Z.$\text{ where n}\in \text{Z}.$
• tan x = tan y implies x = nπ + y,$\text{tan x }=\text{ tan y implies x }=\text{ n}\pi +\text{ y},$ where n∈Z.$\text{ where n}\in \text{Z}.$
• sin2x=sin2y${\sin }^{2}x={\sin }^{2}y$ gives x=nπ±y$x=n\pi \pm y$
• cos2x=cos2y${\cos }^{2}x={\cos }^{2}y$ gives x=nπ±y$x=n\pi \pm y$
• tan2x=tan2y${\tan }^{2}x={\tan }^{2}y$ gives x=nπ±y