Trigonometry Ratios and Identities - Notes, Concept and All Important Formula

TRIGONOMETRIC RATIOS & IDENTITIES

• 1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES :
• 2. BASIC TRIGONOMETRIC IDENTITIES :
• 3. SIGNS OF TRIGONOMETRIC FUNCTIONS IN DIFFERENT QUADRANTS:
• 4. TRIGONOMETRIC FUNCTIONS OF ALLIED ANGLES :
• 5. IMPORTANT TRIGONOMETRIC FORMULAE :
• 6. VALUES OF SOME T-RATIOS FOR ANGLES $18^{\circ}, 36^{\circ}, 15^{\circ}$, $22.5^{\circ}, 67.5^{\circ}$ etc.
• 7. MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC EXPRESSIONS :
• 8. IMPORTANT RESULTS :
• 9. CONDITIONAL IDENTITIES :
• 10. DOMAINS, RANGES AND PERIODICITY OF TRIGONOMETRIC FUNCTIONS :
• 11. GRAPH OF TRIGONOMETRIC FUNCTIONS
• 12. IMPORTANT NOTE :
• FAQ

1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES :

$\dfrac{D}{90}=\dfrac{G}{100}=\dfrac{2 C}{\pi}$

1 Radian $=\dfrac{180}{\pi}$ degree $\approx 57^{\circ} 17^{\prime} 15^{\prime \prime}$ (approximately)

1 degree $=\dfrac{\pi}{180}$ radian $\approx 0.0175$ radian

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All Chapter Notes, Concept and Important Formula

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2. BASIC TRIGONOMETRIC IDENTITIES :

(a) $\sin ^{2} \theta+\cos ^{2} \theta=1$ or $\sin ^{2} \theta=1-\cos ^{2} \theta$ or $\cos ^{2} \theta=1-\sin ^{2} \theta$
(b) $\sec ^{2} \theta-\tan ^{2} \theta=1$ or $\sec ^{2} \theta=1+\tan ^{2} \theta$ or $\tan ^{2} \theta=\sec ^{2} \theta-1$
(c) If $\sec \theta+\tan \theta$$=\mathrm{k} \Rightarrow \sec \theta-\tan \theta$$=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \sec \theta$$=\mathrm{k}+\dfrac{1}{\mathrm{k}}$
(d) $\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$ or $\operatorname{cosec}^{2} \theta=1+\cot ^{2} \theta$ or $\cot ^{2} \theta=\operatorname{cosec}^{2} \theta-1$
(e) If $\operatorname{cosec} \theta+\cot \theta$$=\mathrm{k} \Rightarrow \operatorname{cosec} \theta-\cot \theta$$=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \operatorname{cosec} \theta$$=\mathrm{k}+\dfrac{1}{\mathrm{k}}$

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3. SIGNS OF TRIGONOMETRIC FUNCTIONS IN DIFFERENT QUADRANTS:

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4. TRIGONOMETRIC FUNCTIONS OF ALLIED ANGLES :

(a) $\sin (2 n \pi+\theta)=\sin \theta, \cos (2 n \pi+\theta)=\cos \theta$, where $n \in \mathbb{I}$

(b) 

• $\sin (-\theta)=-\sin \theta$
• $\cos (-\theta)=\cos \theta$
• $\sin \left(90^{\circ}-\theta\right)=\cos \theta \quad$
• $\cos \left(90^{\circ}-\theta\right)=\sin \theta$
• $\sin \left(90^{\circ}+\theta\right)=\cos \theta \quad$
• $\cos \left(90^{\circ}+\theta\right)=-\sin \theta$
• $\sin \left(180^{\circ}-\theta\right)=\sin \theta$
• $\cos \left(180^{\circ}-\theta\right)=-\cos \theta$
• $\sin \left(180^{\circ}+\theta\right)=-\sin \theta$
• $\cos \left(180^{\circ}+\theta\right)=-\cos \theta$
• $\sin \left(270^{\circ}-\theta\right)=-\cos \theta$
• $\cos \left(270^{\circ}-\theta\right)=-\sin \theta$
• $\sin \left(270^{\circ}+\theta\right)=-\cos \theta$
• $\cos \left(270^{\circ}+\theta\right)=\sin \theta$

Note :
(i) $\sin n \pi=0 ; \cos n \pi=(-1)^{n} ; \tan n \pi=0$, where $n \in I$
(ii) $\sin (2 n+1) \dfrac{\pi}{2}=(-1)^{n} ; \cos (2 n+1) \dfrac{\pi}{2}=0$, where $n \in I$

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5. IMPORTANT TRIGONOMETRIC FORMULAE :

(i) $\sin (A+B)=\sin A \cos B+\cos A \sin B$.

(ii) $\sin (A-B)=\sin A \cos B-\cos A \sin B$.

(iii) $\cos (A+B)=\cos A \cos B-\sin A \sin B$

(iv) $\cos (A-B)=\cos A \cos B+\sin A \sin B$

(v) $\tan (A+B)=\dfrac{\tan A+\tan B}{1-\tan A \tan B}$

(vi) $\tan (\mathrm{A}-\mathrm{B})=\dfrac{\tan \mathrm{A}-\tan \mathrm{B}}{1+\tan \mathrm{A} \tan \mathrm{B}}$

(vii) $\cot (A+B)=\dfrac{\cot B \cot A-1}{\cot B+\cot A}$

(viii) $\cot (A-B)=\dfrac{\cot B \cot A+1}{\cot B-\cot A}$

(ix) $2 \sin A \cos B=\sin (A+B)+\sin (A-B)$.

(x) $2 \cos A \sin B=\sin (A+B)-\sin (A-B)$.

(xi) $2 \cos A \cos B=\cos (A+B)+\cos (A-B)$

(xii) $2 \sin A \sin B=\cos (A-B)-\cos (A+B)$

(xiii) $\sin C+\sin D=2 \sin \left(\dfrac{C+D}{2}\right) \cos \left(\dfrac{C-D}{2}\right)$

(xiv) $\sin C-\sin D=2 \cos \left(\dfrac{C+D}{2}\right) \sin \left(\dfrac{C-D}{2}\right)$

(xv) $\cos C+\cos D=2 \cos \left(\dfrac{\mathrm{C}+\mathrm{D}}{2}\right) \cos \left(\dfrac{\mathrm{C}-\mathrm{D}}{2}\right)$

(xvi) $\cos C-\cos D$$=2 \sin \left(\dfrac{C+D}{2}\right) \sin \left(\dfrac{D-C}{2}\right)$

(xvii) $\sin 2 \theta=2 \sin \theta \cos \theta=\dfrac{2 \tan \theta}{1+\tan ^{2} \theta}$

(xviii) $\cos 2 \theta$$=\cos ^{2} \theta-\sin ^{2} \theta$$=2 \cos ^{2} \theta-1$$=1-2 \sin ^{2} \theta$$=\dfrac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}$

(xix) $1+\cos 2 \theta=2 \cos ^{2} \theta$ or $|\cos \theta|=\sqrt{\dfrac{1+\cos 2 \theta}{2}}$

(xx) $1-\cos 2 \theta$$=2 \sin ^{2} \theta$ or $|\sin \theta|$$=\sqrt{\dfrac{1-\cos 2 \theta}{2}}$

(xxi) $\tan \theta=\dfrac{1-\cos 2 \theta}{\sin 2 \theta}=\dfrac{\sin 2 \theta}{1+\cos 2 \theta}$

(xxii) $\tan 2 \theta$$=\dfrac{2 \tan \theta}{1-\tan ^{2} \theta}$

(xxiii) $\sin 3 \theta=3 \sin \theta-4 \sin ^{3} \theta$.

(xxiv) $\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta$.

(xxv) $\tan 3 \theta=\dfrac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta}$

(xxvi) $\sin ^{2} A-\sin ^{2} B$$=\sin (A+B) \cdot \sin (A-B)$$=\cos ^{2} B-\cos ^{2} A$

(xxvii) $\cos ^{2} A-\sin ^{2} B=\cos (A+B) \cdot \cos (A-B)$.

(xxviii) $\sin (A+B+C)$
$=\sin A \cos B \cos C$$+\sin B \cos A \cos C$$+\sin C \cos A \cos B$$-\sin A \sin B \sin C$
$=\sum \sin A \cos B \cos C-\Pi \sin A$
$=\cos A \cos B \cos C[\tan A+\tan B+\tan C-$$\tan A \tan B \tan C]$

(xxix) $\cos (A+B+C)$
$=\cos A \cos B \cos C$$-\sin A \sin B \cos C$$-\sin A \cos B \sin C$$-\cos A \sin B \sin C$
$=\Pi \cos A-\Sigma \sin A \sin B \cos C$
$=\cos A \cos B \cos C[1-\tan A \tan B-\tan B \tan C$$-\tan C \tan A]$

(xxx) $\tan (\mathrm{A}+\mathrm{B}+\mathrm{C})$

$=\dfrac{\tan A+\tan B+\tan C-\tan A \tan B \tan C}{1-\tan A \tan B-\tan B \tan C-\tan C \tan A}$$=\dfrac{S_{1}-S_{3}}{1-S_{2}}$

(xxxi) $\sin \alpha+\sin (\alpha+\beta)+\sin (\alpha+2 \beta)+\ldots$$... \sin (\alpha+\overline{n-1} \beta)$

$=\dfrac{\sin \left\{\alpha+\left(\dfrac{\mathrm{n}-1}{2}\right) \beta\right\} \sin \left(\dfrac{\mathrm{n} \beta}{2}\right)}{\sin \left(\dfrac{\beta}{2}\right)}$

(xxxii) $\cos \alpha+\cos (\alpha+\beta)+\cos (\alpha+2 \beta)+\ldots$$....+\cos (\alpha+\overline{n-1} \beta)$

$=\dfrac{\cos \left\{\alpha+\left(\dfrac{\mathrm{n}-1}{2}\right) \beta\right\} \sin \left(\dfrac{\mathrm{n} \beta}{2}\right)}{\sin \left(\dfrac{\beta}{2}\right)}$

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6. VALUES OF SOME T-RATIOS FOR ANGLES $18^{\circ}, 36^{\circ}, 15^{\circ}$, $22.5^{\circ}, 67.5^{\circ}$ etc.

(a) $\sin 18^{\circ}=\dfrac{\sqrt{5}-1}{4}=\cos 72^{\circ}=\sin \dfrac{\pi}{10}$
(b) $\cos 36^{\circ}=\dfrac{\sqrt{5}+1}{4}=\sin 54^{\circ}=\cos \dfrac{\pi}{5}$
(c) $\sin 15^{\circ}=\dfrac{\sqrt{3}-1}{2 \sqrt{2}}=\cos 75^{\circ}=\sin \dfrac{\pi}{12}$
(d) $\cos 15^{\circ}=\dfrac{\sqrt{3}+1}{2 \sqrt{2}}=\sin 75^{\circ}=\cos \dfrac{\pi}{12}$
(e) $\tan \dfrac{\pi}{12}=2-\sqrt{3}=\dfrac{\sqrt{3}-1}{\sqrt{3}+1}=\cot \dfrac{5 \pi}{12}$
(f) $\tan \dfrac{5 \pi}{12}=2+\sqrt{3}=\dfrac{\sqrt{3}+1}{\sqrt{3}-1}=\cot \dfrac{\pi}{12}$
(g) $\tan \left(22.5^{\circ}\right)=\sqrt{2}-1=\cot \left(67.5^{\circ}\right)$$=\cot \dfrac{3 \pi}{8}=\tan \dfrac{\pi}{8}$
(h) $\tan \left(67.5^{\circ}\right)=\sqrt{2}+1=\cot \left(22.5^{\circ}\right)$

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7. MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC EXPRESSIONS :

(a) a $\cos \theta+\mathrm{b} \sin \theta$ will always lie in the interval $\left[-\sqrt{a^{2}+b^{2}}, \sqrt{a^{2}+b^{2}}\right]$, i.e. the maximum and minimum values are $\sqrt{a^{2}+b^{2}},-\sqrt{a^{2}+b^{2}}$ respectively.

(b) Minimum value of $a^{2} \tan ^{2} \theta+b^{2} \cot ^{2} \theta=2 a b$, where $a, b>0$

(c) Minimum value of $a^{2} \cos ^{2} \theta+b^{2} \sec ^{2} \theta$ (or $\left.a^{2} \sin ^{2} \theta+b^{2} \operatorname{cosec}^{2} \theta\right)$ is either $2 \mathrm{ab}$ (when $|\mathrm{a}| \geq|\mathrm{b}|$ ) or $\mathrm{a}^{2}+\mathrm{b}^{2}$ (when $|\mathrm{a}| \leq|\mathrm{b}|$ ).

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8. IMPORTANT RESULTS :

(a) $\sin \theta \sin \left(60^{\circ}-\theta\right) \sin \left(60^{\circ}+\theta\right)=\dfrac{1}{4} \sin 3 \theta$
(b) $\cos \theta \cdot \cos \left(60^{\circ}-\theta\right) \cos \left(60^{\circ}+\theta\right)=\dfrac{1}{4} \cos 3 \theta$
(c) $\tan \theta \tan \left(60^{\circ}-\theta\right) \tan \left(60^{\circ}+\theta\right)=\tan 3 \theta$
(d) $\cot \theta \cot \left(60^{\circ}-\theta\right) \cot \left(60^{\circ}+\theta\right)=\cot 3 \theta$
(e) (i) $\sin ^{2} \theta+\sin ^{2}\left(60^{\circ}+\theta\right)+\sin ^{2}\left(60^{\circ}-\theta\right)=\dfrac{3}{2}$
    (ii) $\cos ^{2} \theta+\cos ^{2}\left(60^{\circ}+\theta\right)+\cos ^{2}\left(60^{\circ}-\theta\right)=\dfrac{3}{2}$
(f) (i) If $\tan A+\tan B+\tan C=\tan A \tan B \tan C$
    then $\mathrm{A}+\mathrm{B}+\mathrm{C}=\mathrm{n} \pi, \mathrm{n} \in \mathrm{I}$
   (ii) If $\tan A \tan B+\tan B \tan C+\tan C \tan A=1$
    then $A+B+C=(2 n+1) \dfrac{\pi}{2}, n \in I$
(g) $\cos \theta \cos 2 \theta \cos 4 \theta \ldots . \cos \left(2^{\mathrm{n}-1} \theta\right)=\dfrac{\sin \left(2^{\mathrm{n}} \theta\right)}{2^{\mathrm{n}} \sin \theta}$
(h) $\cot A-\tan A=2 \cot 2 A$

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9. CONDITIONAL IDENTITIES :

If $A+B+C=180^{\circ}$, then
(a) $\tan A+\tan B+\tan C=\tan A \tan B \tan C$
(b) $\cot A \cot B+\cot B \cot C+\cot C \cot A=1$
(c) $\tan \dfrac{A}{2} \tan \dfrac{B}{2}+\tan \dfrac{B}{2} \tan \dfrac{C}{2}+\tan \dfrac{C}{2} \tan \dfrac{A}{2}$$=1$
(d) $\cot \dfrac{A}{2}+\cot \dfrac{B}{2}+\cot \dfrac{C}{2}=\cot \dfrac{A}{2} \cot \dfrac{B}{2} \cot \dfrac{C}{2}$
(e) $\sin 2 A+\sin 2 B+\sin 2 C=4 \sin A \sin B \sin C$
(f) $\cos 2 A+\cos 2 B+\cos 2 C$$=-1-4 \cos A \cos B \cos C$
(g) $\sin A+\sin B+\sin C=4 \cos \dfrac{A}{2} \cos \dfrac{B}{2} \cos \dfrac{C}{2}$
(h) $\cos A+\cos B+\cos C$$=1+4 \sin \dfrac{A}{2} \sin \dfrac{B}{2} \sin \dfrac{C}{2}$

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10. DOMAINS, RANGES AND PERIODICITY OF TRIGONOMETRIC FUNCTIONS :

$\begin{array}{|l|l|l|c|} \hline & \text { Domain } & \text { Range } & \text { Period } \\ \hline \sin \mathrm{x} & \mathrm{R} & {[-1,1]} & 2 \pi \\ \hline \cos \mathrm{x} & \mathrm{R} & {[-1,1]} & 2 \pi \\ \hline \tan \mathrm{x} & \mathrm{R}-((2 \mathrm{n}+1) \pi / 2 ; \mathrm{n} \in \mathrm{I}\} & \mathrm{R} & \pi \\ \hline \cot \mathrm{x} & \mathrm{R}-[\mathrm{n} \pi: \mathrm{n} \in \mathrm{I}] & \mathrm{R} & \pi \\ \hline \sec \mathrm{x} & \mathrm{R}-\{(2 \mathrm{n}+1) \pi / 2: \mathrm{n} \in \mathrm{I}\} & (-\infty,-1] \cup[1, \infty) & 2 \pi \\ \hline \operatorname{cosec} \mathrm{x} & \mathrm{R}-\{\mathrm{n} \pi: \mathrm{n} \in \mathrm{I}\} & (-\infty,-1] \cup[1, \infty) & 2 \pi \\ \hline \end{array}$

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11. GRAPH OF TRIGONOMETRIC FUNCTIONS

y = sinx

y = cosx

y = tanx

y = cosecx

y = secx

y = cotx

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12. IMPORTANT NOTE :

(a) The sum of interior angles of a polygon of $\mathrm{n}$ -sides $=(n-2) \times 180^{\circ}=(\mathrm{n}-2) \pi$
(b) Each interior angle of a regular polygon of $\mathrm{n}$ sides $=\dfrac{(\mathrm{n}-2)}{\mathrm{n}} \times 180^{\circ}=\dfrac{(\mathrm{n}-2)}{\mathrm{n}} \pi$
(c) Sum of exterior angles of a polygon of any number of sides $=360^{\circ}=2 \pi$

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FAQ

What are the basic Trigonometry ratios?

sinθ, tanθ, cosθ, secθ, cotθ and cosecθ

What are the formula of Trigonometry Ratios ?

1. sinθ= (side opposite of angle θ )/ (Hypotenuse)

 2. cosθ=(side adjacent to angle θ)/(Hypotenuse) 

3. tanθ=(side opposite of angle θ )/(side adjacent to angle θ) 

4. cosecθ= reciprocal of sinθ

5. cotθ= reciprocal of tanθ

6. secθ= reciprocal of cosθ

What are the fundamental Identities of trigonometry?

1. sin²θ+cos²θ=1

2. 1+cot²θ=cosec²θ

3. 1+tan²θ=sec²θ

How many ratios are there in Trigonometry?

There are only 6 Trigonometry Ratios. sinθ, tanθ, cosθ, secθ, cotθ and cosecθ

Mathematics-Important Notes, concept & Formula