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)
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Copyright 1996 © Cuthbert A. Nyack.