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

1. English System/Degree system

1 right-angle = 90°

1° = 60' (60 minute)

1' = 60'' (60 second)

1 right-angle = 100g (100 gradian)

1g = 100' (100 minute)

1' = 100'' (100 second)

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.)

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

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

πC = 180°

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

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

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

En = Euler Number

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

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