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

TRIGONOMETRIC RATIOS & IDENTITIES

    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


    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 \(\operatorname{cosec} \theta+\cot \theta\)\(=\mathrm{k} \Rightarrow \operatorname{cosec} \theta-\cot \theta\)\(=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \operatorname{cosec} \theta\)\(=\mathrm{k}+\dfrac{1}{\mathrm{k}}\)


    3. SIGNS OF TRIGONOMETRIC FUNCTIONS IN DIFFERENT QUADRANTS:

    SIGNS OF TRIGONOMETRIC FUNCTIONS IN DIFFERENT QUADRANTS


    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 \(n \in I\)


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

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


    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) \(\tan \left(67.5^{\circ}\right)=\sqrt{2}+1=\cot \left(22.5^{\circ}\right)\)


    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 \(|\mathrm{a}| \leq|\mathrm{b}|\) ).


    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) \(\cot A-\tan A=2 \cot 2 A\)


    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) \(\cos A+\cos B+\cos C\)\(=1+4 \sin \dfrac{A}{2} \sin \dfrac{B}{2} \sin \dfrac{C}{2}\)


    10. DOMAINS, RANGES AND PERIODICITY OF TRIGONOMETRIC FUNCTIONS :

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


    11. GRAPH OF TRIGONOMETRIC FUNCTIONS

    y = sinx

    Trigonometry function graph of sinx

    y = cosx

    Trigonometry function graph of cosx

    y = tanx

    Trigonometry function graph of tanx

    y = cosecx
    Trigonometry function graph of cosecx

    y = secx

    Trigonometry function graph of secx

    y = cotx

    Trigonometry function graph of cotx


    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 \(=360^{\circ}=2 \pi\)

    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

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