# TRIG

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```					mTRIG DERIVS                                        sin x sin y = ½ (cos(x – y) – cos (x + y))
sin x = cos x                                       cos x cos y = ½ (cos(x – y) + cos (x + y))
cos x = -sin x                                      sin x cos y = ½ (sin(x + y) + sin(x – y))
tan x = sec2x
sec x = sec x * tan x                               EXPONENT RULES
csc x = -csc x * cot x                              x0 = 1
cot x = -csc2x                                      xm * xn = xm+n
sin-1x = 1 / (√(1 – x2))                            (xm)n = x(m*n)
tan-1x = 1 / (1 + x2)                               (x ± y)2 = x2 ± 2xy + y2

DERIVS                                              LOG RULES
xn = nxn-1                                          Key to remember: logs are exponents!
√x = 1/2 * x-1/2                                    logac = b ≡ ab = c
3
√x = 1/3 * x-2/3                                  logc(ab) = logca + logcb
enx = nenx                                          logc(a/b) = logca - logcb
cx = ln(c) * cx                                     logc(an) = n * logca
ln x = 1/x                                          logbb = 1
logcx = 1 / x * ln c                                logc(n√a) = logc(a1/n) = logc(a) / n
(f(x) + g(x))' = f'(x) + g'(x)                      logc1 = 0
(f(x) * g(x))' = f'(x)*g(x) + f(x)*g'(x)            logc(1/a) = logc1 – logca = -logca
(f(x) / g(x))' = f'(x)*g(x) – f(x)*g'(x)
(g(x))2                    L’HOPITAL
(f(g(x)))' = f'(g(x)) * g'(x)                       If lim x->a results in 0/0 or ∞/∞
f(x)/g(x) = f’(x)/g’(x)
TRIG – These are NOT derivatives
sinΘ = opp / hyp                                    BASIC LOGIC OF SHORTEST DISTANCE
cosΘ = adj / hyp                                    1. Solve for y 2. Plug in to x2 + y2
3. Simplify and derive, set = 0
4. Solve for x 5. Plug x into step 1, solve for y
1
X                                 PARAMETRIC CURVES
x = f(t) and y = g(t) used to get point P
Speed of P at t = √(f’(t)2 + g’(t)2)
√(1 – x2)                                     Slope (derivative) = g’(t) / f’(t)
sinΘ from bottom left                               Circle: x(t) = sin(tπ) y(t) = cos(tπ)

sin2x + cos2x = 1                                   BASIC LOGIC OF RATE OF CHANGE
sec2x = 1 + tan2x                                   Known rate = x’(t) Desired = s’(t)
sin(2x) = 2 sin x * cos x                           Derive an equation that uses x(t) and s(t)
cos(2x) = cos2x - sin2x = 2cos2x - 1 = 1 - 2sin2x   Such as s(t)2 = x(t)2 + 102 (a triangle)
sin2x = ½ (1 - cos(2x))                             Plug in know rate x’(t), and solve for s’(t)
cos2x = ½ (1 + cos(2x))
sin(x ± y) = sin x * cos y ± sin y * cos x
cos(x ± y) = cos x * cos y (-+) sin x * sin y
tan(x ± y) = tan x ± tan y
1 (-+) tan x * tan y
INTEGRATION                                            ∫ √(p2 - x2) =
Odd functions (to an odd power) integrated in          ½ (x√(p2 – x2) + p2 arcsin(x/p)), p > 0
the range [-a,a] are = 0
Example: sin(x), x5, tan(x), (x3 + x)                  EXPRESSIONS CONTAINING TRIG
∫ab F’(x) = F(b) – F(a)                                ∫ sin2(ax) dx = x/2 – sin(2ax)/4a
Integrating between two functions:                     ∫ sinn(ax) dx = -sinn-1(ax)cos(ax)/na +
∫ab (f(x) – g(x))                                                      n-1/n ∫ sinn-2(ax) dx , n > 0
2
∫ cos (ax) dx = x/2 + sin(2ax)/4a
ESTIMATION                                             ∫ cosn(ax) dx = cosn-1(ax) sin(ax) / na +
Ln = ∑ni=1 f(xn-1)∆x                                                   n-1/n ∫ cosn-2(ax) dx
Rn = ∑ni=1 f(xn)∆x                                     ∫ x sin(ax) dx = 1/a2 sin(ax) – x/a cos(ax)
Mn = ∑ni=1 f((xn-1+xn)/2)∆x                            ∫ x cos(ax) dx = 1/a2 cos(ax) + x/a sin(ax)
Tn = ∑ni=1 ((f(xn-1) + f(xn))/2)∆x
DISK / SLICE RULE
BASIC FORMS                                            Area of rotation: slice perpendicular to axis of
∫ xn dx = xx+1 / n+1 , n != -1                         rotation – if conical, use πr2 where r = y
∫ x-1 dx = ln |x|                                      Integrate over bounds of x axis
∫ ex dx = ex
∫ bx dx = bx / ln b                                    SHELL RULE
∫ sin x dx = - cos x                                   For rotation about y axis
∫ cos x dx = sin x                                     2πxy
∫ tan x dx = ln | sec x | = -ln | cos x |              or 2πx(y1 – y2) for hollow object
∫ cot x dx = ln | sin x | = -ln | csc x |              Integrate over bounds of x axis
∫ sec2x dx = tan x                                     OR
∫ csc2x dx = -cot x                                    πr2 where r = x – treat as disk rule
∫ sec x tan x dx = sec x
∫ csc x cot x dx = -csc x                              LENGTH OF CURVE
∫ 1 / (x2 + a2) dx = arctan(x/a) / a , a != 0            1. y = f(x)
∫ 1 / (x2 - a2) dx = 1/2a ln | x-a / x+a |               2. find f’(x)
∫ 1 / √(a2 – x2) dx = arcsin(x/a) , a > 0                3. find ds/dx = √(1 + (dy/dx)2)
∫ ln x dx = x(ln x – 1)                                  4. Integrate ds/dx from a to b
∫ sec x dx = ln | sec x + tan x | = ln | tan (x/2 +
π/4) |                                                 RATES
∫ sec x dx = ln | csc x – cot x | = ln | tan (x/2) |   PV = P0ert
∫ csc x dx = ln | csc x – cot x | = ln |tan(x/2) |     Continuous
∫ 1 / (a + bepx) dx = x/a – 1/ap ln | a + bepx |       FV = ert ∫ab P0e-rt

EXPRESSIONS CONTAINING ax + b
∫ (ax + b)n dx = (ax + b)n+1 / a(n + 1) , n != -1
∫ 1 / (ax + b) = 1/a ln | ax + b |
∫ x / (ax + b) = x/a – b/a2 ln | ax + b |
∫ x / (ax + b)2 dx = b/a2(ax+b) + 1/a2 ln | ax+b |

EXPRESSIONS CONTAINING ax2 + c
∫ √(x2 ± p2) =
½ (x√(x2 ± p2) ± p2 ln | x + √(x2 ± p2)|)

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