All trigonometry functions and Identity or formulas list | pdf

Trigonometric function

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\(sin\hspace{1mm}θ = \frac{p}{h}\)    -1 ≤ sin θ ≤ 1

\(cos\hspace{1mm}θ = \frac{b}{h}\)     -1 ≤ cos θ ≤ 1

\(tan\hspace{1mm}θ = \frac{p}{b}\)     -∞ ≤ tan θ ≤ ∞

\(cot\hspace{1mm}θ = \frac{b}{p}\)     -∞ ≤ cot θ ≤ ∞

\(sec\hspace{1mm}θ = \frac{h}{b}\)     -1 ≥ sec θ ≥ 1

\(cosec\hspace{1mm}θ = \frac{h}{p}\)    -1 ≥ cosec θ ≥ 1

Method Of measuring the angles

1. English System/Degree system

2. French System/Gradian system

3. Circular System/Radian system

1. English System/Degree system

      1 right-angle = 90°

      1° = 60' (60 minute) 

      1' = 60'' (60 second)

2. French System/Gradian system

      1 right-angle = 100^g (100 gradian)

      1^g = 100' (100 minute) 

      1' = 100'' (100 second)

3. Circular System/Radian system

        1 right-angle = 90° = \(\frac{π^C}{2} \left ( \frac{π}{2} \hspace{1mm} Radian \right )\)

\(1^C = \left ( \frac{180\times 7}{22} \right )^{\circ} = 57^{\circ} 16' 22''\)  (approx.)

Relation Between Degree, Gradian and Radian

\[ 90^{\circ} = 100^g = \left ( \frac{π}{2} \right )^C\]

\[\frac{D }{90} = \frac{G}{100} = \frac{2R }{π}\]

π^C = 180°

Degree	Radian
30°	\(\frac{\pi}{6}\)
45°	\(\frac{\pi}{4}\)
60°	\(\frac{\pi}{3}\)
90°	\(\frac{\pi}{2}\)
180°	\(\pi\)
270°	\(\frac{3\pi}{2}\)
360°	\(2\pi\)

Trigonometric Identity or formulas

cos (0° - θ) = cos (-θ) = cos θ

sec (0° - θ) = sec (-θ) = sec θ

sin (0° - θ) = sin (-θ) = -sin θ

cosec (0° - θ) = cosec (-θ) = cosec θ

tan (0° - θ) = tan (-θ) = -tan θ

cot (0° - θ) = cot (-θ) = -cot θ

sin \(\frac{A}{2}=\pm \sqrt{\frac{1-cos A}{2}}\)

cos \(\frac{A}{2}=\pm \sqrt{\frac{1+cos A}{2}}\)

tan \(\frac{A}{2}=\pm \sqrt{\frac{1-cos A}{1+cos A}}\)

sin θ cosec θ = 1

cos θ sec θ = 1

tan θ cot θ = 1

sin² θ + cos² θ = 1 

⇒ sin² θ = 1 - cos² θ 

⇒ cos² θ = 1 - sin² θ

sec² θ - tan² θ = 1

⇒ sec θ - tan θ = \(\frac {1}{sec θ + tan θ}\)

cosec² θ - cot² θ = 1

⇒ cosec θ - cot θ = \(\frac {1}{cosec θ + cot θ}\)

sin (A + B) = sin A cos B + cos A sin B

sin (A - B) = sin A cos B - cos A sin B

cos (A + B) = cos A cos B - sin A sin B

cos (A - B) = cos A cos B + sin A sin B

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

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

2 cos A cos B = cos (A + B) + cos (A - B)

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

sin 2θ = 2 sin θ cos θ

sinθ = \(2 sin \frac {θ}{2} cos \frac {θ}{2}\)

sin 2θ = \(\frac {2tanθ}{1 - tan^2 θ}\)

cos 2θ = cos² θ - sin² θ

cos 2θ = 1 - 2sin² θ

cos 2θ = 2cos² θ - 1

cos θ = \(cos^2\frac {θ}{2} - sin^2\frac {θ}{2}\)

cos θ = \(1 - 2sin^2 \frac {θ}{2}\)

cos θ = \(2cos^2 \frac {θ}{2} - 1\)

cos 2θ =\( \frac{1 - tan^2 θ} {1 + tan^2 θ}\)

sin 3θ = 3sin θ - 4sin³ θ

sin θ = \(3 sin \frac {θ}{3} - 4 sin^3 \frac {θ}{3}\)

cos 3θ = 4cos³ θ - 3 cos θ

cos θ = \(4 cos^3 \frac {θ}{3} - 3 cos \frac {θ}{3}\)

sin θ sin 2θ sin 4θ = \(\frac {1}{4}\) sin 3θ cos θ cos 2θ cos 4θ = \(\frac {1}{4}\) cos 3θ tan θ tan 2θ tan 4θ = tan 3θ

tan 3θ  = \(\frac {3tan θ - tan^3θ}{1 - 3tan^2θ}\)

tan θ  = \(\frac {3tan \frac {θ}{3}- tan^3\frac {θ}{3}}{1 - 3tan^2\frac {θ}{3}}\)

sin 3θ = 4 sin (60° - θ) sin θ sin (60° + θ)

cos 3θ = 4 cos (60° - θ) cos θ cos (60° + θ)

tan 3θ = tan (60° - θ) tan θ tan (60° + θ)

3tan 3θ = tan θ + tan (60° + θ) + tan (120° + θ) 

sin (A + B) sin (A - B) = sin² A - sin² B

sin (A + B) sin (A - B) = cos² B - cos² A

sin (A + B + C) = sin A.cos B.cos C + cos A Sin B cos C + Cos A Cos B Sin C - Sin A sin B sin C

cos (A + B) cos (A - B) = cos² A - sin² B

cos (A + B) cos (A - B) = cos² B - sin² A

cos (A + B + C) = cos A cos B cos C - cos A sin B sin C - sin A cos B sin C - sin A sin B cos C

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

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

cot (A + B) = \(\frac { cot A cot B - 1}{cot A + cot B}\)

cot (A - B) = \(\frac { cot A cot B + 1}{cot A - cot B}\)

tan \((\frac {A-B}{2} = \sqrt {\frac {1 - cos (A-B)}{1 + cos (A+B)}}\)

tan (A  + B + C) = \(\frac{tan A + tan B +  tan C - tan A.tan B.tan C}{1-tan A.tan B-tan B.tan C-tan C.tanA}\)

If (1 + tan A)(1 + tan B) = 2 then, A + B = 45°

\(\frac {1 + cos θ}{1 - cos θ} = cot^2\frac {θ}{2}\)

\(\frac {1 - cos θ}{1 + cos θ} = tan^2\frac {θ}{2}\)

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

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

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

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

1 + sin 2θ = (sin θ + cos θ)²

1 - sin 2θ = (sin θ - cos θ)²

cot θ + tan θ = 2cosec 2θ

cot θ - tan θ = 2cot 2θ

tan θ.sec 2θ = tan 2θ - tan θ 

tan² θ tan 2θ = tan 2θ - 2tan θ

cos θ cos 2θ cos (2² θ) cos (2³ θ) ..............cos (2ⁿ θ) = \(\frac {sin (2^{n+1}θ)}{2^{n+1}sin θ}\)

cos θ cos 2θ cos (2² θ) cos (2³ θ) ..............cos (2^{n - 1} θ) = \(\frac {sin (2^{n}θ)}{2^{n}sin θ}\)

cos α + cos (α + β) + cos (α + 2β) + ............n terms = \(\frac {sin\hspace{1mm} n\frac{β}{2}}{sin \frac{β}{2}} cos \left (\frac {first \hspace{1mm} angle + last \hspace{1mm} angle}{2}\right )\)

Some Important values

sin 75° = cos 15° = \(\frac {\sqrt{3} + 1}{2\sqrt{2}}\)

cos 75° = sin 15° = \(\frac {\sqrt{3} - 1}{2\sqrt{2}}\)

tan 75° = cot 15° = \(\frac {\sqrt{3} + 1}{\sqrt{3} - 1} = 2 + \sqrt{3}\)

cot 75° = tan 15° = \(\frac {\sqrt{3} - 1}{\sqrt{3} + 1} = 2- \sqrt{3}\)

\(sin 67\tfrac {1°}{2} = cos 22\tfrac {1°}{2} = \frac{1}{2} \sqrt{2+\sqrt{2}}\)

\(cos 67\tfrac {1°}{2} = sin 22\tfrac {1°}{2} = \frac{1}{2} \sqrt{2-\sqrt{2}}\)

\(tan 67\tfrac {1°}{2} = cot 22\tfrac {1°}{2} = \sqrt{2}+1\)

\(cot 67\tfrac {1°}{2} = tan 22\tfrac {1°}{2} = \sqrt{2}-1\)

sin 18° = cos 72° = \(\frac{\sqrt{5}-1}{4}\)

cos 36° = sin 54° = \(\frac{\sqrt{5}+1}{4}\)

cos 18° = sin 72° = \(\frac{\sqrt{10+2\sqrt{5}}}{4}\)

sin 36° = cos 54° = \(\frac{\sqrt{10-2\sqrt{5}}}{4}\)

General solution of Trigonometric equations

Trigonometric equation	General solution
sin θ = sin α	θ = nπ + (-1)nα [n ∈ I or n ∈ Z]
cos θ = cos α	θ = 2nπ ± α [n ∈ I or n ∈ Z]
tan θ = tan α	θ = nπ + α [n ∈ I or n ∈ Z]
sin θ = 0 sin θ = sin 0°	θ = nπ [n ∈ I or n ∈ Z]
cos θ = 0 cos θ = cos \(\frac{π}{2}\)	θ = 2nπ ± \(\frac{π}{2}\) or θ = (2n + 1) \(\frac{π}{2}\) or θ = (2n - 1) \(\frac{π}{2}\) [n ∈ I or n ∈ Z]
tan θ = 0 tan θ = tan 0°	θ = nπ [n ∈ I or n ∈ Z]
sin θ = 1 sin θ = sin \(\frac{π}{2}\)	θ = nπ + (-1)n\(\frac{π}{2}\) or θ = (4n + 1)\(\frac{π}{2}\) [n ∈ I or n ∈ Z]
cos θ = 1 cos θ = cos 0°	θ = 2nπ [n ∈ I or n ∈ Z]
tan θ = 1 tan θ = tan \(\frac{π}{4}\)	θ = nπ + \(\frac{π}{4}\) [n ∈ I or n ∈ Z]
sin θ = -1	θ = (4n - 1) \(\frac{π}{2}\) [n ∈ I or n ∈ Z]
cos θ = -1	θ = (2n - 1)π or θ = (2n + 1)π [n ∈ I or n ∈ Z]
tan θ = -1 or tan θ = tan \(\frac{-π}{4}\) or tan θ = tan \(\frac{3π}{4}\)	θ = nπ - \((\frac{-π}{4})\) or θ = nπ + \(\frac{3π}{4}\) [n ∈ I or n ∈ Z]
sin2 θ = sin2 α	θ = nπ ± α [n ∈ I or n ∈ Z]
cos2 θ = cos2 α	θ = nπ ± α [n ∈ I or n ∈ Z]
tan2 θ = tan2 α	θ = nπ ± α [n ∈ I or n ∈ Z]
sin θ = sin α and cos θ = cos α	θ = 2nπ + α [n ∈ I or n ∈ Z]
sin θ = sin α and tan θ = tan α	θ = 2nπ + α [n ∈ I or n ∈ Z]
cos θ = cos α and tan θ = tan α	θ = 2nπ + α [n ∈ I or n ∈ Z]

 Formulas and Properties of Triangle

Half perimeter of the triangle S\( = \frac {a+b+c}{2}\)

Area of triangle △ = \( \frac {1}{2} \times Base \times Height\)

Area of triangle △ = \( \sqrt{S(S-a)(S-b)(S-c)} \)

△ = \(  \frac {1}{2} \times ab \times Sin \hspace{1mm} C \)

△ = \(  \frac {1}{2} \times bc \times Sin \hspace{1mm} A \)

△ = \(  \frac {1}{2} \times ac \times Sin \hspace{1mm} B \)

Where a,b,c are length of triangle sides and A,B,C are angles of triangle.

1. Sine Formula

\( \frac {a}{Sin \hspace{1mm} A} = \frac {b}{Sin \hspace{1mm} B} = \frac {c}{Sin \hspace{1mm} C} \)

\( \frac {Sin \hspace{1mm} A}{a} = \frac {Sin \hspace{1mm} B}{b} = \frac {Sin \hspace{1mm} C}{c} \)

\( \frac {a}{Sin \hspace{1mm} A} = \frac {b}{Sin \hspace{1mm} B} = \frac {c}{Sin \hspace{1mm} C} = K \)

2. Cosine formula

Cos A = \(\frac {b^2+c^2-a^2}{2bc}\)

Cos B = \(\frac {a^2+c^2-b^2}{2ac}\)

Cos C = \(\frac {a^2+b^2-c^2}{2ab}\)

3. Projection formula

a = b cos C + c cos B

b = a cos C + c cos A

c = a cos B + b cos A

4. Tangent formula (Napier's formula)

\(tan \left ( \frac {A-B}{2} \right ) = \left ( \frac {a-b}{a+b} \right ) cot \frac {C}{2} \)

\(tan \left ( \frac {B-C}{2} \right ) = \left ( \frac {b-c}{b+c} \right ) cot \frac {A}{2} \)

\(tan \left ( \frac {C-A}{2} \right ) = \left ( \frac {c-a}{c+a} \right ) cot \frac {B}{2} \)

5. Half angle formula

sin \( \frac {A}{2} = \sqrt {\frac {(S-b)(S-c)}{bc}} \)

sin \( \frac {B}{2} = \sqrt {\frac {(S-a)(S-c)}{ac}} \)

sin \( \frac {C}{2} = \sqrt {\frac {(S-a)(S-b)}{ab}} \)

cos \( \frac {A}{2} = \sqrt {\frac {S(S-a)}{bc}} \)

cos \( \frac {B}{2} = \sqrt {\frac {S(S-b)}{ac}} \)

cos \( \frac {C}{2} = \sqrt {\frac {S(S-c)}{ab}} \)

tan \( \frac {A}{2} = \sqrt {\frac {(S-b)(S-c)}{S(S-a)}} \)

tan \( \frac {B}{2} = \sqrt {\frac {(S-a)(S-c)}{S(S-b)}} \)

tan \( \frac {C}{2} = \sqrt {\frac {(S-a)(S-b)}{S(S-c)}} \)

tan \( \frac {A}{2} = \sqrt {\frac {△}{S(S-a)}} \)

tan \( \frac {B}{2} = \sqrt {\frac {△}{S(S-b)}} \)

tan \( \frac {C}{2} = \sqrt {\frac {△}{S(S-c)}} \)

Some other important formula in triangle

In-radius (r) = \( \frac {△}{S} \)

Circum radius (R) = \( \frac {abc}{4△} \)

\( \frac {a}{Sin \hspace{1mm} A} = \frac {b}{Sin \hspace{1mm} B} = \frac {c}{Sin \hspace{1mm} C} = 2R \)

\( \frac {a}{Sin \hspace{1mm} A} = \frac {b}{Sin \hspace{1mm} B} = \frac {c}{Sin \hspace{1mm} C} = \frac {abc}{2△} \)

r = (S - a) tan \(\frac {A}{2}\)

r = (S - a) tan \(\frac {B}{2}\)

r = (S - a) tan \(\frac {C}{2}\)

ex-radius r₁ = \( \frac {△}{S-a} \)

ex-radius r₂ = \( \frac {△}{S-b} \)

ex-radius r₃ = \( \frac {△}{S-c} \)

In Triangle ABC :-

sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C

cos 2A + cos 2B + cos 2C = -1 - 4 cos A cos B cos C

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

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

sin² A + sin² B + sin² C = 2 + 2 cos A cos B cos C

cos² A + cos² B + cos² C = 1 - 2 cos A cos B cos C

tan A + tan B + tan C = tan A tan B tan C

cot A cot B + cot B cot C + cot C cot A = 1

Trigonometric functions series or progressions 

This series is also called Taylor's expansions. 

sin x = \(x - \frac {x^3}{3!} + \frac {x^5}{5!} - \frac {x^7}{7!} + .......\)

cos x = \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ......\)

\(tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \frac{62x^9}{2835} +......+\frac {2^{2n}(2^{2n} - 1)B_nx^{2n-1}}{(2n)!}+ .....\)

B_n = Bernoulli Number

\(cosec x = \frac{1}{x} + \frac{x}{6} + \frac{7x^3}{360} + \frac{31x^5}{15120} +......+\frac {2(2^{2n} - 1)B_nx^{2n-1}}{(2n)!}+ .....\)

sec x = \(1 + \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720} + ...... + \frac{E_nx^{2n}}{(2n)!} + ....\)

E_n = Euler Number

cot x = \(\frac{1}{x} - \frac{x}{3} - \frac{x^3}{45} - \frac{2x^5}{945} -......-\frac {2^{2n}B_nx^{2n-1}}{(2n)!}+ .....\)

If you need any formula for trigonometry. you can tell it in the comment box. I will soon provide it.

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