Spread the love

Trigonometry & Hyperbolic Functions – Complete Formula Sheet

Angle Units Standard Trig Values Signs in Quadrants Allied Angles (90°, 180°, 270°, 360°) Addition & Subtraction Sum ↔ Product Multiple & Sub-multiple Half-Angle & Power-Reduction Area of Triangle & Radii Inverse Trigonometric Functions Principal Values More Exact Angles (15°, 18°, 36°, 54°, 72°, 75°, 22.5°) Hyperbolic Functions Relations: Circular ↔ Hyperbolic Periodic Functions (Periods)

Angle Units

\(1^\circ = 60′ = 3600”\), \(1\ \text{right angle} = 90^\circ = \tfrac{\pi}{2}\ \text{radians}\).

\(\pi\ \text{radians} = 180^\circ\), \(2\ \text{right angles} = \pi\ \text{radians} = 180^\circ\).

WhatsApp Group Join Now

Telegram Group Join Now

Standard Trigonometric Values (0°–90°)

Angle	0°	30°	45°	60°	90°
sin	0	\(\tfrac12\)	\(\tfrac{1}{\sqrt2}\)	\(\tfrac{\sqrt3}{2}\)	1
cos	1	\(\tfrac{\sqrt3}{2}\)	\(\tfrac{1}{\sqrt2}\)	\(\tfrac12\)	0
tan	0	\(\tfrac{1}{\sqrt3}\)	1	\(\sqrt3\)	Not defined

Signs in Different Quadrants

I: All + II: sin, cosec + III: tan, cot + IV: cos, sec +.

Allied Angles (Reduction Formulas)

Form	Equivalent
\(\sin(90^\circ-\theta)\)	\(\cos\theta\)
\(\sin(90^\circ+\theta)\)	\(\cos\theta\)
\(\cos(90^\circ-\theta)\)	\(\sin\theta\)
\(\cos(90^\circ+\theta)\)	\(-\sin\theta\)
\(\tan(90^\circ-\theta)\)	\(\cot\theta\)
\(\tan(90^\circ+\theta)\)	\(-\cot\theta\)
\(\sin(180^\circ-\theta)\)	\(\sin\theta\)
\(\sin(180^\circ+\theta)\)	\(-\sin\theta\)
\(\cos(180^\circ-\theta)\)	\(-\cos\theta\)
\(\cos(180^\circ+\theta)\)	\(-\cos\theta\)
\(\tan(180^\circ-\theta)\)	\(-\tan\theta\)
\(\tan(180^\circ+\theta)\)	\(\tan\theta\)
\(\sin(270^\circ\pm\theta)\)	\(-\cos\theta\)
\(\cos(270^\circ+\theta)\)	\(\sin\theta\)
\(\cos(270^\circ-\theta)\)	\(-\sin\theta\)
\(\sin(360^\circ-\theta)\)	\(-\sin\theta\)
\(\cos(360^\circ-\theta)\)	\(\cos\theta\)

Addition & Subtraction Formulae

\(\sin(A\pm B)=\sin A\cos B \pm \cos A\sin B\)

\(\cos(A\pm B)=\cos A\cos B \mp \sin A\sin B\)

\(\tan(A\pm B)=\dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)

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

\(\sin(A+B)\sin(A-B)=\sin^2A-\sin^2B\)

\(\cos(A+B)\cos(A-B)=\cos^2A-\sin^2B\)

Sum ↔ Product Identities

\(\sin C+\sin D=2\sin\dfrac{C+D}{2}\cos\dfrac{C-D}{2}\)

\(\sin C-\sin D=2\cos\dfrac{C+D}{2}\sin\dfrac{C-D}{2}\)

\(\cos C+\cos D=2\cos\dfrac{C+D}{2}\cos\dfrac{C-D}{2}\)

\(\cos C-\cos D=-2\sin\dfrac{C+D}{2}\sin\dfrac{C-D}{2}\)

\(\sin C\sin D=\dfrac{\cos(C-D)-\cos(C+D)}{2}\)

\(\cos C\cos D=\dfrac{\cos(C-D)+\cos(C+D)}{2}\)

\(\sin C\cos D=\dfrac{\sin(C+D)+\sin(C-D)}{2}\)

Multiple & Sub-multiple Angles

Double Angle

\(\sin2A=2\sin A\cos A\)

\(\cos2A=\cos^2A-\sin^2A=1-2\sin^2A=2\cos^2A-1\)

\(\tan2A=\dfrac{2\tan A}{1-\tan^2A}\)

Triple Angle

\(\sin3A=3\sin A-4\sin^3A\)

\(\cos3A=4\cos^3A-3\cos A\)

\(\tan3A=\dfrac{3\tan A-\tan^3A}{1-3\tan^2A}\)

Half-Angle & Power-Reduction

\(\sin^2\dfrac{A}{2}=\dfrac{1-\cos A}{2}\), \(\cos^2\dfrac{A}{2}=\dfrac{1+\cos A}{2}\), \(\tan^2\dfrac{A}{2}=\dfrac{1-\cos A}{1+\cos A}=\dfrac{\sin A}{1+\cos A}=\dfrac{1-\cos A}{\sin A}\)

इन्हीं से \(\sin\dfrac{A}{2}, \cos\dfrac{A}{2}, \tan\dfrac{A}{2}\) के sign quadrant से तय करें।

Area of Triangle, Circumradius \(R\) & Inradius \(r\)

\(\Delta=\dfrac12 ab\sin C=\dfrac12 bc\sin A=\dfrac12 ca\sin B\)

Heron: \(s=\dfrac{a+b+c}{2}\), \(\Delta=\sqrt{s(s-a)(s-b)(s-c)}\)

Circumradius: \(R=\dfrac{a}{2\sin A}=\dfrac{b}{2\sin B}=\dfrac{c}{2\sin C}=\dfrac{abc}{4\Delta}\)

Inradius: \(r=\dfrac{\Delta}{s}\), Exradii: \(r_a=\dfrac{\Delta}{s-a}\), \(r_b=\dfrac{\Delta}{s-b}\), \(r_c=\dfrac{\Delta}{s-c}\)

Inverse Trigonometric Functions (Principal Branch)

Function	Domain	Range
\(\sin^{-1}x\)	\([-1,1]\)	\(\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\)
\(\cos^{-1}x\)	\([-1,1]\)	\([0,\pi]\)
\(\tan^{-1}x\)	\((-\infty,\infty)\)	\(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\)
\(\cot^{-1}x\)	\((-\infty,\infty)\)	\((0,\pi)\)
\(\sec^{-1}x\)	\((-\infty,-1]\cup[1,\infty)\)	\([0,\pi]\setminus\{\tfrac{\pi}{2}\}\)
\(\csc^{-1}x\)	\((-\infty,-1]\cup[1,\infty)\)	\(\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\setminus\{0\}\)

Principal Values (Examples)

Expression	Principal Value
\(\sin^{-1}\!\big(\sin \tfrac{2\pi}{3}\big)\)	\(\tfrac{\pi}{3}\)
\(\sin^{-1}\!\big(\sin \tfrac{4\pi}{3}\big)\)	\(-\tfrac{\pi}{3}\)
\(\cos^{-1}\!\big(\cos \tfrac{7\pi}{6}\big)\)	\(\tfrac{5\pi}{6}\)
\(\cos^{-1}\!\big(\cos \tfrac{5\pi}{3}\big)\)	\(\tfrac{\pi}{3}\)
\(\tan^{-1}\!\big(\tan \tfrac{3\pi}{4}\big)\)	\(-\tfrac{\pi}{4}\)
\(\tan^{-1}\!\big(\tan \tfrac{7\pi}{4}\big)\)	\(-\tfrac{\pi}{4}\)

Trigonometric Ratios of Some More Angles

नीचे दी गई मान्यताएँ exact surds के साथ दी गई हैं।

Angle

sin

cos

tan

cot

15°

\(\tfrac{\sqrt6-\sqrt2}{4}\)

\(\tfrac{\sqrt6+\sqrt2}{4}\)

\(2-\sqrt3\)

\(2+\sqrt3\)

75°

\(\tfrac{\sqrt6+\sqrt2}{4}\)

\(\tfrac{\sqrt6-\sqrt2}{4}\)

\(2+\sqrt3\)

\(2-\sqrt3\)

18°

\(\tfrac{\sqrt5-1}{4}\sqrt{2+\phi}\)

Hyperbolic Functions (Definitions & Identities)

\(\sinh z=\dfrac{e^{z}-e^{-z}}{2}\), \(\cosh z=\dfrac{e^{z}+e^{-z}}{2}\), \(\tanh z=\dfrac{\sinh z}{\cosh z}\)

Key Relations

\(\cosh^2 z-\sinh^2 z=1\)

\(\operatorname{sech}^2 z=1-\tanh^2 z\), \(\operatorname{csch}^2 z=\coth^2 z-1\)

Angle Addition

\(\sinh(z_1\pm z_2)=\sinh z_1\cosh z_2\pm\cosh z_1\sinh z_2\)

\(\cosh(z_1\pm z_2)=\cosh z_1\cosh z_2\pm\sinh z_1\sinh z_2\)

\(\tanh(z_1\pm z_2)=\dfrac{\tanh z_1\pm\tanh z_2}{1\pm\tanh z_1\tanh z_2}\)

Relations between Circular & Hyperbolic

\(\sin(iz)=i\sinh z\), \(\cos(iz)=\cosh z\), \(\tan(iz)=i\tanh z\)

Periodic Functions (Periods)

Function	Fundamental Period
\(\sin x, \cos x, \sec x, \csc x\)	\(2\pi\)
\(\tan x, \cot x\)	\(\pi\)
\(\|\sin x\|\)	\(\pi\)
\(\sin^2 x, \cos^2 x\)	\(\pi\)
\(\sin(ax+b)\)	\(\dfrac{2\pi}{\|a\|}\)
\(a\cos bx+b\sin x\) (same \(b\))	\(\dfrac{2\pi}{\|b\|}\)

Trigonometry Formula Sheet (Compact)

Angle Units & Signs

\(1^\circ=60′,\ 1’=60”;\ \pi\ \text{rad}=180^\circ\).

Quadrants: I: All +, II: sin,cosec +, III: tan,cot +, IV: cos,sec +.

Standard Values (0°–90°)

θ	0°	30°	45°	60°	90°
sin	0	\(\tfrac12\)	\(\tfrac1{\sqrt2}\)	\(\tfrac{\sqrt3}{2}\)	1
cos	1	\(\tfrac{\sqrt3}{2}\)	\(\tfrac1{\sqrt2}\)	\(\tfrac12\)	0
tan	0	\(\tfrac1{\sqrt3}\)	1	\(\sqrt3\)	NA

Allied / Reduction

\(\sin(90^\circ\!-\!θ)=\cos θ\)	\(\cos(90^\circ\!-\!θ)=\sin θ\)
\(\sin(180^\circ\!-\!θ)=\sin θ\)	\(\cos(180^\circ\!-\!θ)=-\cos θ\)
\(\tan(180^\circ\!-\!θ)=-\tan θ\)	\(\tan(90^\circ\!\pm\!θ)=\mp\cot θ\)
\(\sin(360^\circ\!-\!θ)=-\sin θ\)	\(\cos(360^\circ\!-\!θ)=\cos θ\)

Addition / Subtraction

\(\sin(A\!\pm\!B)=\sin A\cos B\pm\cos A\sin B\)

\(\cos(A\!\pm\!B)=\cos A\cos B\mp\sin A\sin B\)

\(\tan(A\!\pm\!B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)

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

Sum ↔ Product

\(\sin C+\sin D=2\sin\frac{C+D}{2}\cos\frac{C-D}{2}\)

\(\cos C+\cos D=2\cos\frac{C+D}{2}\cos\frac{C-D}{2}\)

\(\cos C-\cos D=-2\sin\frac{C+D}{2}\sin\frac{C-D}{2}\)

\(\sin C\sin D=\frac{\cos(C-D)-\cos(C+D)}{2}\)

Double / Triple / Half

\(\sin2A=2\sin A\cos A\)

\(\cos2A=1-2\sin^2A=2\cos^2A-1\)

\(\tan2A=\dfrac{2\tan A}{1-\tan^2A}\)

\(\sin3A=3\sin A-4\sin^3A\), \(\cos3A=4\cos^3A-3\cos A\)

\(\tan3A=\dfrac{3\tan A-\tan^3A}{1-3\tan^2A}\)

\(\sin^2\frac{A}{2}=\dfrac{1-\cos A}{2}\), \(\ \cos^2\frac{A}{2}=\dfrac{1+\cos A}{2}\)

\(\tan^2\frac{A}{2}=\dfrac{1-\cos A}{1+\cos A}=\dfrac{\sin A}{1+\cos A}\)

Triangle: Area & Radii

\(\Delta=\tfrac12 ab\sin C=\tfrac12 bc\sin A=\tfrac12 ca\sin B\)

Heron: \(s=\tfrac{a+b+c}{2}\), \(\ \Delta=\sqrt{s(s-a)(s-b)(s-c)}\)

Circumradius: \(R=\dfrac{abc}{4\Delta}=\dfrac{a}{2\sin A}=\dfrac{b}{2\sin B}=\dfrac{c}{2\sin C}\)

Inradius: \(r=\dfrac{\Delta}{s}\); \(\ r_a=\dfrac{\Delta}{s-a}\), \(r_b=\dfrac{\Delta}{s-b}\), \(r_c=\dfrac{\Delta}{s-c}\)

Inverse Trig (Principal)

f	Domain	Range
\(\sin^{-1}x\)	\([-1,1]\)	\(\big[-\tfrac{\pi}{2},\tfrac{\pi}{2}\big]\)
\(\cos^{-1}x\)	\([-1,1]\)	\([0,\pi]\)
\(\tan^{-1}x\)	\(\mathbb R\)	\(\big(-\tfrac{\pi}{2},\tfrac{\pi}{2}\big)\)
\(\cot^{-1}x\)	\(\mathbb R\)	\((0,\pi)\)

Periods

\(\sin x,\cos x,\sec x,\csc x\)	\(2\pi\)
\(\tan x,\cot x\)	\(\pi\)
\(\sin^2x,\cos^2x,\|\sin x\|\)	\(\pi\)
\(\sin(ax+b)\)	\(\dfrac{2\pi}{\|a\|}\)

More Angles (15°, 22.5°, 72°…), Hyperbolic & Relations

Exact Angles

\(\tan 15^\circ=2-\sqrt3,\ \tan 22.5^\circ=\sqrt2-1,\ \tan 75^\circ=2+\sqrt3\).

\(\sin22.5^\circ=\dfrac{\sqrt{2-\sqrt2}}{2},\ \cos22.5^\circ=\dfrac{\sqrt{2+\sqrt2}}{2}\).

Hyperbolic (Quick)

\(\sinh z=\frac{e^z-e^{-z}}{2},\ \cosh z=\frac{e^z+e^{-z}}{2},\ \tanh z=\frac{\sinh z}{\cosh z}\)

\(\cosh^2z-\sinh^2z=1,\ \tanh^2z+\operatorname{sech}^2z=1\).

\(\sin(iz)=i\sinh z,\ \cos(iz)=\cosh z,\ \tan(iz)=i\tanh z\).

Tip: प्रिंट करने से पहले ब्राउज़र स्केल 90–95% रखें तो एक पेज में समा जाता है।

Advanced Trigonometric & Hyperbolic Formulas

Special Angles (15°, 18°, 22.5°, 36°, 54°, 72°, 75°)

\(\tan15^\circ = 2-\sqrt3,\quad \cot15^\circ=2+\sqrt3\)

\(\tan75^\circ = 2+\sqrt3,\quad \cot75^\circ=2-\sqrt3\)

\(\tan22.5^\circ=\sqrt2-1,\quad \cot22.5^\circ=\sqrt2+1\)

\(\sin22.5^\circ=\tfrac{\sqrt{2-\sqrt2}}{2},\quad \cos22.5^\circ=\tfrac{\sqrt{2+\sqrt2}}{2}\)

18°, 36°, 54°, 72° के लिए golden-ratio surds (√5 वाली identities) उपयोग करें।

Circular Functions of Complex Variable

\(\cos z=\dfrac{e^{iz}+e^{-iz}}{2},\quad \sin z=\dfrac{e^{iz}-e^{-iz}}{2i}\)

\(\tan z=\dfrac{e^{iz}-e^{-iz}}{i(e^{iz}+e^{-iz})}\)

Hyperbolic Functions

\(\sinh z=\tfrac{e^z-e^{-z}}{2},\quad \cosh z=\tfrac{e^z+e^{-z}}{2},\quad \tanh z=\dfrac{\sinh z}{\cosh z}\)

Identities: \(\cosh^2z-\sinh^2z=1\), \(\sech^2z=1-\tanh^2z\), \(\csch^2z=\coth^2z-1\)

Relations: Circular ↔ Hyperbolic

\(\sin(iz)=i\sinh z,\quad \cos(iz)=\cosh z,\quad \tan(iz)=i\tanh z\)

\(\sinh(iz)=i\sin z,\quad \cosh(iz)=\cos z\)

Multiple-Angle (Hyperbolic)

\(\sinh2z=2\sinh z\cosh z,\quad \cosh2z=\cosh^2z+\sinh^2z\)

\(\tanh2z=\tfrac{2\tanh z}{1+\tanh^2z}\)

\(\sinh3z=3\sinh z+4\sinh^3z,\quad \cosh3z=4\cosh^3z-3\cosh z\)

Important Results

\(\cos0\cos2^\circ\cos4^\circ…\cos88^\circ=\dfrac{\sin90^\circ}{2^{45}}=\dfrac{1}{2^{44}}\)

\(\sin α+\sin(α+β)+\sin(α+2β)+…=\dfrac{\sin(\tfrac{nβ}{2})\sin(α+\tfrac{n-1}{2}β)}{\sin(\tfrac{β}{2})}\)

\(\cos α+\cos(α+β)+…=\dfrac{\sin(\tfrac{nβ}{2})\cos(α+\tfrac{n-1}{2}β)}{\sin(\tfrac{β}{2})}\)

Periodic Functions

Function	Period
\(\sin x, \cos x, \sec x, \csc x\)	\(2\pi\)
\(\tan x, \cot x\)	\(\pi\)
\(\sin^2x, \cos^2x, \|\sin x\|\)	\(\pi\)
\(\sin(ax+b)\)	\(\tfrac{2\pi}{\|a\|}\)
\(a\cos bx+b\sin x\) (same freq)	\(\tfrac{2\pi}{\|b\|}\)

यह हिस्सा तीसरे पेज के लिए है। इसे पहले वाले HTML के नीचे paste कर दीजिए।

Shekhar

लेखक परिचय – चंद्रशेखर
मैं चंद्र शेखर, एक प्रशिक्षित और समर्पित गणित शिक्षक हूं। मैं MadhyamikPariksha.com का संस्थापक हूं। मेरा उद्देश्य छात्रों को सही, सरल और भरोसेमंद शैक्षिक सामग्री उपलब्ध कराना है।

मेरी शैक्षणिक योग्यता इस प्रकार है:

🎓 M.Sc (गणित)
📘 B.Ed
🔬 B.Sc (PCM)
✅ TGT Qualified (Maths) – 2016
📝 UP TET Qualified

मुझे गणित पढ़ाने का 7 वर्षों का अनुभव है। मैंने हजारों छात्रों को बोर्ड परीक्षाओं और प्रतियोगी परीक्षाओं की तैयारी में मार्गदर्शन दिया है। मेरी खासियत है – गणित को आसान भाषा और रोचक तरीके से समझाना।
वेबसाइट के बारे में
MadhyamikPariksha.com एक निजी शैक्षिक पोर्टल है, जहाँ छात्र हिंदी माध्यम में पढ़ाई से जुड़ी उपयोगी सामग्री पा सकते हैं। यहां उपलब्ध हैं:

माध्यमिक और उच्च माध्यमिक परीक्षाओं की तैयारी सामग्री
2. पुराने प्रश्न पत्र और हल
3.गणित क्विज़, मॉक टेस्ट, और अपडेट्स

सरकारी पोर्टल नहीं है
स्पष्टीकरण: यह वेबसाइट सरकारी पोर्टल नहीं है। इसका किसी भी सरकारी विभाग, बोर्ड या संस्था से कोई संबंध नहीं है। यह एक निजी प्रयास है, जिसका मकसद छात्रों को मदद पहुंचाना है।
हमारा उद्देश्य
हमारा लक्ष्य है कि हर छात्र को पढ़ाई में मार्गदर्शन मिले, चाहे वह बोर्ड परीक्षा की तैयारी कर रहा हो या प्रतियोगी परीक्षा की। हम विषयों को आसान भाषा में, बिना डर के समझाने में यकीन रखते हैं।

अगर आपको कोई सुझाव या प्रश्न हो, तो आप संपर्क करें पेज के माध्यम से मुझसे जुड़ सकते हैं।
चंद्रशेखर
(M.Sc Maths, B. Sc, B.Ed, TGT Qualified 2016, UPTET Qualified)