Trigonometry

sin ²x + cos ²x = 1

1 + cot ²x = csc ²x

tan ²x + 1 = sec ²x

ENGELSHON'S EQUATION

a/c = sin A/ sin C    b/c = sin B/sin C

Theorem Engelshon's Equation 

(a+b)/c = (sin A + sin B)/ sin C     

(a-b)/c = (sin A - sin B)/sin C

THE LAW OF COSINE

a² = b² + c² – 2bc.cos A

AREA OF A TRIANGLE

A= 1/2 a.b.sin C

or

A= (s(s-a)(s-b)(s-c))^1/2

Inscribed Circle

r = (((s-a)(s-b)(s-c)/s))^1/2

Circumscribed Circle

r =  abc/(4(s(s-a)(s-b)(s-c))^1/2)


ANGLE-ADDITION AND ANGLE SUBSTRACTION

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

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

tan (A + B) = (tan A + tan B)/ (1 - tan A. tan B)
tan (A - B) = (tan A - tan B)/ (1 + tan A. tan B)



DOUBLE ANGLES FORMULAS

derived cos 2A within cos (A + B) formula and subtitute B with A so that:

cos (A + A)  = cos A.cos A - sin A.sinA
cos 2A = cos ²A -  sin ²A

if  you subtitute sin ²A with (1- cos ²A) or  cos ²A with (1- sin ²A)

cos 2A = 2cos ²A -  1
or
cos 2A = 1 -  2sin ²A

the same  goes for tangent and sine formulas

so that:

sin 2A = 2 sin A. cos A

and

tan 2A = 2 tan A/ (1-  tan ²A)


THE PRODUCT FORMULAS

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)

THE SUM AND DIFFERENCE FORMULA

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

THEOREM


y = a sin x + b cos x = (a ² + b ²)^1/2sin (x + s)