Trigonometry Formulas and Calculus Formulas

Peter Y. Woo, Biola Univ., 8/27/99

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(np/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 (7p/2-x) = -sin x because it is in the 3rd quadrant, assuming 0 < x < p/2. E.g., sin (p-x) = sin x, sin (p+x) = sin x, cos (-x) = cos x, sin (-x) = -sin x.
sin² + cos² = 1, \ sin = Ö(1-cos²), cos = Ö(1-sin²).
tan² + 1 = sec², \ tan = Ö(sec²-1), sec = Ö(1+tan²).
cot² + 1 = csc², \ cot = Ö(csc²-1), csc = Ö(1+cot²).
Rare formulas: sin = tan/Ö(1 + tan²), cos = 1/Ö(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 ¹ 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 = (½)b c sin A = (½)c a sin B = (½)a b sin C = Ö[ 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.
sin (A/2) = Ö[(1-cos A)/2], cos (A/2) = Ö[(1+cos A)/2],
tan (A/2) = Ö[^{(1-cos A)}/_{(1 +cos A)}] = ^{(1-cos A)}/_{sin A} = ^{sin A}/_{(1+cos A)}
tan (p/4+A/2) = Ö[^{(1+sin A)}/_{(1-sin A)}] = ^{(cos A/2 + sin A/2)} / _{(cos A/2-sin A/2)}
= ^{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 = p/2.

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 + Ö(1 + x²)),
ch^-1x = ln (x + Ö(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 + Ö(1 - x²))}/x| = ch^-1(1/x)
csch^-1x = ln |^{(1 + Ö(1 + x²))}/x| = sh^-1(1/x)
sh x + ch x = e^x, ch x - sh x = e^-x

    Calculus Formulas. D denotes ^d/_dx
D c = 0, D x = 1, D (1/x) = -1/x², D xⁿ = n x^n-1. D Öx = 1 / 2Öx.
ò 1 dx = x.     ò x dx = x²/2.     ò xⁿ dx = xⁿ⁺¹/(n+1).
ò (1/x²) dx = -1/x.     ò (1/Öx) dx = 2Öx.     ò Öx dx = (2/3)x^3/2.
ò (1/x) dx = ln | x |.     D ln x = 1/x.     log_ab = ^{ln b}/_{ln a}.     log_a (a^x) = x = a^log_ax.
D_xlog_ax = 1/ (x ln a).     D e^x = e^x.     D_xa^x = a^x ln a.
ò e^x dx = e^x.     ò a^x dx = a^x / ln a.
D sin x = cos x.     ò cos x dx = sin x.
D cos x = - sin x.     ò sin x dx = - cos x.
D tan x = sec² x.     ò tan x dx = ln sec x.
D cot x = -csc² x.     ò cot x dx = ln sin x.
D sec x = sec x tan x.     ò sec x dx = ln (sec x + tan x) = ln tan (p/4 + x/2).
D csc x = -csc x cot x.     ò csc x dx = ln (csc x-cot x) = ln tan (x/2).
ò sin²x dx = (½)(x - sin x cos x).     ò cos²x dx = (½)(x + sin x cos x).
ò tan²x dx = tan x -x.     ò cot²x dx = -cot x -x.
ò sec²x = tan x.     ò csc²x = -cot x.     ò sec x tan x = sec x.     ò csc x cot x = -csc x.
D sin^-1x = 1/Ö(1-x²).     D cos^-1x = -1/Ö(1-x²).
D tan^-1x = 1/(1 + x²).     D cot^-1x = -1/(1 + x²).
D sec^-1x = 1/[xÖ(x²-1)].     D csc^-1x = -1/[xÖ(x²-1)].
ò (1/Ö(1-x²)) dx = sin^-1x.     ò (1/Ö(a²-x²)) dx = sin^-1(x/a).
ò (1+x²)^-1 dx = tan^-1x.     ò (a²+x²)^-1 dx = (1/a) tan^-1(x/a).
ò (x Ö(x²-1))^-1 dx = sec^-1x.     ò (x Ö(x²-a²))^-1 dx = (1/a) sec^-1(x/a).
ò Ö(a²-x²) dx = (½)a²sin^-1(x/a) + (½)xÖ(a²-x²).
ò sec³x dx = (½)(sec x tan x + ln |sec x + tan x|).

D sh x = ch x.     ò sh x dx = ch x.     D ch x = sh x.     ò ch x dx = sh x.
D th x = sech²x.     ò th x dx = ln ch x.     D cth x = -csch²x.     ò cth x = ln sh x.
D sech x = -sech x th x.     ò sech x dx = tan^-1sh x (Wow!)
D csch x = -csch x cth x.     ò csch x dx = ln th (x/2) = ln (cth x -csch x).
ò sh²x dx = (½)(sh x ch x -x).     ò ch²x dx = (½)(sh x ch x + x)
ò sech²x dx = th x.     ò csch²x dx = -cth x.
ò sech x th x dx = -sech x.     ò csch x cth x dx = -csch x.
D sh^-1x = 1/Ö(1+x²).     ò (a²+x²)^-1/2 dx = sh^-1(x/a) = ln |x + Ö(a²+x²)|.
D ch^-1x = 1/Ö(x²-1).     ò (x²-a²)^-1/2 dx = ch^-1(x/a) = ln |x + Ö(x²-a²)|.
D th^-1x = 1/(1-x²).     ò (a²-x²)^-1 dx = (1/a) th^-1 (x/a) = (1/2a) ln |(a+x)/(a-x)|.
D cth^-1x = 1/(1-x²).     ò (x²-a²)^-1 dx = -(1/a) th^-1 (x/a) = -(1/2a) ln |(a+x)/(x-a)|.
D sech^-1x = -(x Ö(1-x²))^-1.     D csch^-1x = -(x Ö(1+x²))^-1.
ò (x Ö(a²-x²))^-1 dx = -(1/a) sech^-1(x/a) = -(1/a) ln |(a²+Ö(a²-x²))/ x|
ò (x Ö(a²+x²))^-1 dx = -(1/a) csch^-1(x/a) = -(1/a) ln |(a²+Ö(a²+x²))/ x|
ò Ö(a²+x²) dx = (½)xÖ(a²+x²) + (½)a² sh^-1(x/a).
ò Ö(x²-a²) dx = (½)xÖ(x²-a²) - (½)a² ch^-1(x/a).
ò x e^x dx = e^x (x-1).     ò ln x dx = x (ln x -1).
ò sin^mx cosⁿx dx = (m + n)^-1 [-sin^m-1x cosⁿ⁺¹x + (m-1) ò sin^m-2x cosⁿx dx]
= (m + n)^-1 [sin^m+1x cos^n-1x + (n-1) ò sin^mx cos^n-2x dx]
ò secⁿx dx = (n-1)^-1 [sec^n-2x tan x + (n-2) ò sec^n-2x dx ].
ò tanⁿx dx = tan^n-1x /(n-1) -ò tan^n-2x dx.
ò e^{a x} sin b x dx = (a²+b²)^-1 e^{a x}(a sin b x -b cos b x).
ò e^{a x} cos b x dx = (a²+b²)^-1 e^{a 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: Ö[ (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 | / Ö (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 a from (x, y) to (x', y'):
x' = x cos a + y sin a. y' = y cos a - x sin a.
x = x' cos a - y' sin a. y = y' cos a + x' sin a.
    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 < 2p.
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 ¹ 0, we can rotate axes to make x'y' term zero.

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