# Trigonometric Relations

Cuthbert Nyack

## Everything you wanted to know about trigonometry but were afraid to ask

cos q = cos(-q) = sin(p/2 - q) = cosh jq = ½(ejq + e-jq)
sin q = - sin(-q) = cos(p/2 - q) = - j sinh q = (ejq - e-jq)/2j
tan q = - tan(-q) = cot(p/2 - q) = -j tanh jq = ½(ejq - e-jq)/ j(ejq + e-jq)

For small q:-
sin q = q - q3/3! + q5/5! - ...
cos q = 1 - q2/2! + q4/4! - ...
tan q = q + q3/3 + 2q5/15 + ...
atan q = q - q3/3 + q5/5 + ...

sin 2q = 2 cos q sin q
cos 2q = cos2 q - sin2 q = 2 cos2 q - 1 = 1 - 2 sin2 q
tan 2q = 2 tan q/ (1 - tan2q)

sin q/2 = ± Ö[(1 - cos q)/2]
cos q/2 = ± Ö[(1 + cos q)/2]
tan q/2 = sin q/(1 + cos q)

sin 3q = 3 sin q - 4 sin3 q
cos 3q = - 3 cos q + 4 cos3 q
tan 3q = (3 tan q - tan3 q)/ (1 - 3 tan2q)

cos2 q + sin2 q = 1
sec2 q - tan2 q = 1
cosec2 q - cot2 q = 1

sin2 q = ½(1 - cos 2q)
cos2 q = ½(1 + cos 2q)
tan2 q = (1 - cos 2q) /(1 + cos 2q)

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

sin A + sin B = 2 sin ½(A + B) cos ½(A - B)
sin A - sin B = 2 cos ½(A + B) sin ½(A - B)
cos A + cos B = 2 cos ½(A + B) cos ½(A - B)
cos A - cos B = - 2 sin ½(A + B) sin ½(A - B)
tan A + tan B = sin (A + B)/(cos A cos B)
tan A - tan B = sin (A - B)/(cos A cos B)

sin2A + sin2B = 1 - cos(A + B)cos(A - B)
sin2A - sin2B = sin(A + B)sin(A - B)
cos2A + sin2B = 1 - sin(A + B)sin(A - B)
cos2A - sin2B = cos(A + B)cos(A - B)
cos2A + cos2B = 1 + cos(A + B)cos(A - B)
cos2A - cos2B = - sin(A + B)sin(A - B)

a sin q + b cos q = [Ö(a2 + b2)] sin(q + tan-1(b/a)) = [Ö(a2 + b2)] cos(q - tan-1(a/b))
a sin q - b cos q = [Ö(a2 + b2)] sin(q - tan-1(b/a)) = - [Ö(a2 + b2)] cos(q + tan-1(a/b))

d/dx (sin ax) = a cos ax
d/dx (cos ax) = -a sin ax
d/dx (tan ax) = a sec2 ax

d/dx (1/sin x) = - cos x/sin2 x
d/dx (1/cos x) = sin x/cos2 x

d/dx (sin2 ax) = a sin 2ax
d/dx (cos2 ax) = - a sin 2ax

d/dx (sin-1(x/a)) = 1/ Ö(a2 - x2)
d/dx (cos-1(x/a)) = - 1/ Ö(a2 - x2)
d/dx (tan-1(x/a)) = a/ (a2 + x2)

ò (sin ax) dx = -(1/a) cos ax
ò (cos ax) dx = (1/a) sin ax
ò (tan ax) dx = -(1/a) loge cos ax

ò (sin2 ax) dx = x/2 - (1/4a) sin 2ax
ò (cos2 ax) dx = x/2 + (1/4a) sin 2ax

ò (sinn ax) dx = [(sinn - 1ax cos ax)/na] + [(n - 1)/n] ò (sinn - 2 ax) dx
ò (cosn ax) dx = [(cosn - 1ax sin ax)/na] + [(n - 1)/n] ò (cosn - 2 ax) dx

ò (sin ax cos ax) dx = (1/2a) sin2 ax
ò (sin m(ax) cos m(ax)) dx = (1/2a(m-n)) cos(m-n)ax - (1/2a(m+n)) cos(m+n)ax assuming m2 ne n2
ò (sin2 ax cos2 ax) dx = (1/8)(x - (1/4a)sin 4ax)

ò (sin ax cosm ax) dx = - (1/a(m+1)) cosm+1 (ax)
ò (sinm ax cos ax) dx = (1/a(m+1)) sinm+1 (ax)

ò (x sin ax) dx = - (x/a) cos ax + a-2 sin ax
ò (x cos ax) dx = (x/a) sin ax + a-2 cos ax

ò (x2 sin ax) dx = - (x2 a-1) cos ax + 2 x a-2 sin ax + (2 a-3) cos ax
ò (x2 cos ax) dx = - (x2 a-1) sin ax - 2 x a-2 cos ax + (2 a-3) sin ax

ò (eax sin bx) dx = (a eax sin bx - b eax cos bx)/(a2 + b2)
ò (eax cos bx) dx = (a eax cos bx + b eax sin bx)/(a2 + b2)

For a triangle with sides a, b, c, and angles A, B, C opposite sides a, b and c respectively, the following relations hold.
a2 = b2 + c2 - 2 b c cos A
a/sin A = b/sin B = c/sin C.
(a - b)/(a + b) = tan ½(A - B)/ tan ½(A + B)