Introduction. Formulas are tools in your toolbox. The
more tools you have, the more advantage over others will you have in
competitions and exams. These are the formulas I have memorized and they
benefit me all my life, and made me a professor of mathematics. I am
indebted to Mr Terry Chamberlain, who taught me most of it in my high
school, Queen Elizabeth School, in Hong Kong, during 1956 to 1958.
You should memorize them a bit at a time,
and their proofs.
How to interpret the diagrams:
(1) Product of 2 non-neighbors (nbrs) = middle guy, e.g.,
sin × csc = 1, cos × tan = sin, sech × cth = csch.
(2) Quotient of any function by a nbr = the other nbr, e.g.,
1/sin = csc, sin/cos = tan, sech/th = csch, th/sh = sech.
(3) Sum of squares of top corners of each triangle = square of
lower corner, e.g., sin2 + cos2 = 1, sh2 + 1 = ch2.
Trigonometry:
tan = sin/cos. cot = cos/sin. sec = 1/cos. csc = 1/sin.
CAST-rule: 4th quadrant: only Cos and sec > 0.
1st quadrant: All functions > 0.
2nd quadrant: only Sin and csc > 0.
3rd quadrant: only Tan and cot > 0.
For a formula such as f(nπ/2 ± x), if n is odd
integer, then change f to its co-function. Then the sign is taken from
CAST rule. E.g., cos (7π/2-x) = -sin x because it is in the
3rd quadrant, assuming 0 < x < π/2. E.g., sin (π-x) = sin x,
sin (π+x) = sin x, cos (-x) = cos x, sin (-x) = -sin x.
sin2 + cos2 = 1, ∴ sin = √(1-cos2), cos = √(1-sin2).
tan2 + 1 = sec2, ∴ tan = √(sec2-1), sec = √(1+tan2).
cot2 + 1 = csc2, ∴ cot = √(csc2-1), csc = √(1+cot2).
Rare formulas: sin = tan/√(1 + tan2), cos = 1/√(1 + tan2).
Compound Angles: (12 line poem to be chanted at night).
sin (A+B) = sin A cos B + cos A sin B. ("sin sum equals sin cos plus cos sin")
sin (A-B) = sin A cos B - cos A sin B ("sin dif equals sin cos minus cos sin")
cos (A+B) = cos A cos B - sin A sin B (Watch the signs!)
cos (A-B) = cos A cos B + sin A sin B.
(Learn all proofs using a diagram).
2 sin A cos B = sin (A+B) + sin (A-B) ("two sin cos equals sin sum plus sin dif")
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) (Watch out!)
sin A + sin B = 2 sin (½)(A+B) cos (½)(A-B) ("sin plus sin equals
two sin half sum cos half dif")
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) ("cos minus cos
equals MINUS two sin half sum sin half dif")
Double and Triple Angles.
sin 2A = 2 sin A cos A
cos 2A = cos2A-sin2A = 2 cos2A-1 = 1-2 sin2A. But cos 2A
≠ cos2A+sin2A, why?
sin2A = (½)(1-cos 2A), cos2A = (½)(1+cos 2A), tan2A =
(1-cos 2A) / (1+cos 2A) .
sin 2A = 2 tan A / (1 + tan2A),
cos 2A = (1-tan2A)/(1+tan2A).
tan 2A = 2 tan A / (1-tan2A).
tan (A+B) = (tan A + tan B)/(1- tan A tan B),
tan (A-B) = (tan A - tan B)/(1 + tan A tan b).
cot (A+B) = (cot A cot B- 1)/(cot A + cot B),
cot (A-B) = (cot A cot B + 1)/(cot B - cot A)
sin 3A = 3 sin A -4 sin3A, cos 3A = 4 cos3A -3 cos A.
tan 3A = (3 tan A - tan3A)/(1 - 3 tan2A).
Area of triangle = Δ = (½)b c sin A = (½)c a sin B = (½)a b sin C
= √[ s (s-a) (s-b) (s-c)]
Law of sines: 2 R = a / sin A = b / sin B = c / sin C
Law of cosines: a2 = b2+ c2 - 2 b c cos A, cos A =
(b2+c2-a2)/2 b c.
Half Angles. Let α = A/2, β = G/2, γ = C/2, then
sin α = √[(1-cos A)/2],
cos α = √[(1+cos A)/2],
tan α = √[(1-cos A)/(1 +cos A)] = (1-cos A)/sin A
= sin A/(1+cos A)
tan (π/4+A/2) = √[(1+sin A)/(1-sin A)]
= (cos α + sin α) / (cos α-sin α)
= cos A/(1-sin A) = (1+sin A)/cos A
tan-1a + tan-1b = tan-1[
(a + b)/(1- a b)]
sin-1x + cos-1x = π/2.
s = R(sinA+sinB+sinC) = 4R cosα cosβ cosγ
inradius r = Δ/s = R(cosA+cosB+cosC -1) = 4R sinα sinβ sinγ
exradius r1 = Δ/(s-a) = 4R cosα sinβ sinγ
Hyperbolic Functions.
Any trig formula with product of two sines or two tan's or two csc's have to
change the ± sign. Thus:
1 = -sh2A + ch2A, but ch 2A = ch2A + sh2A.
sh (A± B) = sh A ch B ± ch A sh B
ch (A± B) = ch A ch B ± sh A sh B
th (A± B) = (th A± th B)/(1± th A th B)
Memorize the graphs of sh, ch, th, cth, sech, csch, and th-1.
sh-1x = ln (x + √(1 + x2)),
ch-1x = ln (x + √(x2-1)).
th-1x = (½)ln |(1 + x)/(1-x)|, -1 < x < 1
cth-1x = (½)ln |(1 + x)/(x-1)|, -1 > x, or x > 1
sech-1x = ln |(1 + √(1 - x2))/x| = ch-1(1/x)
csch-1x = ln |(1 + √(1 + x2))/x| = sh-1(1/x)
sh x + ch x = ex, ch x - sh x = e-x
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Calculus Formulas. D denotes d/dx
D c = 0, D x = 1, D (1/x) = -1/x2, D xn = n xn-1.
D √x = 1 / 2√x.
∫ 1 dx = x. ∫ x dx = x2/2. ∫ xn dx =
xn+1/(n+1).
∫ (1/x2) dx = -1/x. ∫ (1/√x) dx = 2√x.
∫ √x dx = (2/3)x3/2.
∫ (1/x) dx = ln | x |. D ln x = 1/x.
logab = ln b/ln a. loga (ax) = x = alogax.
Dxlogax = 1/ (x ln a). D ex = ex. Dxax
= ax ln a.
∫ ex dx = ex. ∫ ax dx = ax / ln a.
D sin x = cos x. ∫ cos x dx = sin x.
D cos x = - sin x. ∫ sin x dx = - cos x.
D tan x = sec2 x. ∫ tan x dx = ln sec x.
D cot x = -csc2 x. ∫ cot x dx = ln sin x.
D sec x = sec x tan x. ∫ sec x dx = ln (sec x + tan x) =
ln tan (π/4 + x/2).
D csc x = -csc x cot x. ∫ csc x dx = ln (csc x-cot x) =
ln tan (x/2).
∫ sin2x dx = (½)(x - sin x cos x).
∫ cos2x dx = (½)(x + sin x cos x).
∫ tan2x dx = tan x -x. ∫ cot2x dx = -cot x -x.
∫ sec2x = tan x. ∫ csc2x = -cot x.
∫ sec x tan x = sec x. ∫ csc x cot x = -csc x.
D sin-1x = 1/√(1-x2). D cos-1x = -1/√(1-x2).
D tan-1x = 1/(1 + x2). D cot-1x = -1/(1 + x2).
D sec-1x = 1/[x√(x2-1)]. D csc-1x = -1/[x√(x2-1)].
∫ (1/√(1-x2)) dx = sin-1x.
∫ (1/√(a2-x2)) dx = sin-1(x/a).
∫ (1+x2)-1 dx = tan-1x.
∫ (a2+x2)-1 dx = (1/a) tan-1(x/a).
∫ (x √(x2-1))-1 dx = sec-1x.
∫ (x √(x2-a2))-1 dx = (1/a) sec-1(x/a).
∫ √(a2-x2) dx = (½)a2sin-1(x/a) + (½)x√(a2-x2).
∫ sec3x dx = (½)(sec x tan x + ln |sec x + tan x|).
D sh x = ch x. ∫ sh x dx = ch x. D ch x = sh x.
∫ ch x dx = sh x.
D th x = sech2x. ∫ th x dx = ln ch x. D cth x = -csch2x.
∫ cth x = ln sh x.
D sech x = -sech x th x. ∫ sech x dx = tan-1sh x (Wow!)
D csch x = -csch x cth x. ∫ csch x dx = ln th (x/2) = ln (cth x -csch x).
∫ sh2x dx = (½)(sh x ch x -x). ∫ ch2x dx = (½)(sh x ch x + x)
∫ sech2x dx = th x. ∫ csch2x dx = -cth x.
∫ sech x th x dx = -sech x. ∫ csch x cth x dx = -csch x.
D sh-1x = 1/√(1+x2). ∫ (a2+x2)-1/2 dx
= sh-1(x/a) = ln |x + √(a2+x2)|.
D ch-1x = 1/√(x2-1). ∫ (x2-a2)-1/2 dx
= ch-1(x/a) = ln |x + √(x2-a2)|.
D th-1x = 1/(1-x2). ∫ (a2-x2)-1 dx
= (1/a) th-1 (x/a) = (1/2a) ln |(a+x)/(a-x)|.
D cth-1x = 1/(1-x2). ∫ (x2-a2)-1 dx
= -(1/a) th-1 (x/a) = -(1/2a) ln |(a+x)/(x-a)|.
D sech-1x = -(x √(1-x2))-1.
D csch-1x = -(x √(1+x2))-1.
∫ (x √(a2-x2))-1 dx = -(1/a) sech-1(x/a)
= -(1/a) ln |(a2+√(a2-x2))/ x|
∫ (x √(a2+x2))-1 dx = -(1/a) csch-1(x/a)
= -(1/a) ln |(a2+√(a2+x2))/ x|
∫ √(a2+x2) dx = (½)x√(a2+x2) + (½)a2 sh-1(x/a).
∫ √(x2-a2) dx = (½)x√(x2-a2) - (½)a2 ch-1(x/a).
∫ x ex dx = ex (x-1). ∫ ln x dx = x (ln x -1).
∫ sinmx cosnx dx = (m + n)-1
[-sinm-1x cosn+1x + (m-1) ∫ sinm-2x cosnx dx]
= (m + n)-1
[sinm+1x cosn-1x + (n-1) ∫ sinmx cosn-2x dx]
∫ secnx dx = (n-1)-1 [secn-2x tan x + (n-2)
∫ secn-2x dx ].
∫ tannx dx = tann-1x /(n-1) -∫ tann-2x dx.
∫ ea x sin b x dx = (a2+b2)-1 ea x(a sin b x -b cos b x).
∫ ea x cos b x dx = (a2+b2)-1 ea x(a cos b x + b sin b x).
Coordinate Geometry.
Straight Lines. General form: a x + b y + c = 0.
"Slope + intercept" form: y = m x + c, where m is the slope, c is the intercept on y-axis.
"Point + slope" form: y-y1 = m (x-x1), where (x1, y1) is a
known point on the line, m the slope.
"Two intercepts" form: x/a + y/b = 1, where a, b are intercepts on x-axis
and y-axis.
"Two points" form: (y-y1 / (x-x1) = (y2-y1) / (x2-x1).
where (x1, y1), (x2, y2) are known points on the line.
Distance Formulas.
Between 2 points: √[ (x1-x2)2 + (y1-y2)2 ]
Between a point (x1, y1) and a line a x + b y + c = 0:
| a x1 + b y1 + c | / √ (a2+b2)
Angles.
Angle between two lines y = m1x + c1 and y = m2x + c2 is
tan-1 [(m1-m2)/(1 + m1m2)]. They are ⊥ if m1m2 = -1.
Rotation of axes, through α from (x, y) to (x', y'):
x' = x cos α + y sin α. y' = y cos α - x sin α.
x = x' cos α - y' sin α. y = y' cos α + x' sin α.
Circles.
(x-x1)2 + (y-y1)2 = r2. A typical point is
x = x1 + r cos t, y = y1 + r sin t, for 0 ≤ t < 2π.
Tangent at a point (x0, y0) on the circle is
x x0-(x + x0)x1 + x12 + y y0-(y + y0)y1 + y12 = r2.
Parabola.
If a parabola is y2 = 4 a x, then its vertex is (0,0), focus is (a, 0),
latus rectum is x = a with length 4 a, directrix is x = -a.
A typical point on it is (y2/4 a, y) with y as a parameter, or (a t2, 2 a t)
with t as parameter, called "the point t".
The latus rectum is the chord joining the point t= -1 to the point t=1.
Tangent at the point t is y t = x + a t2 which has slope 1/t and goes
through the point (-a t2, 0) also.
Normal at the point t is y-2 a t = -t(x-a t2).
General Conic.
a x2 + 2 h x y + b y2 + 2 g x + 2 f y + c = 0.
If x y term is missing, then it is either an ellipse or hyperbola, and
if furthermore a = b, then it is circle. If a = -b, it is rectangular
hyperbola. However, they may also be degenerate form, into a point circle
or two str. lines.
If a x2 + 2 h x y + b y2 is a perfect square, = (m x + n y)2,
then it is parabola, or else 2 str. lines if g x + f y = p (m x + n y) for
some p.
If h ≠ 0, we can rotate axes to make x'y' term zero.
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