Precalculus Cheat Sheet
,„ ,
Ztan(a)
§5.1 Polynomial Functions and Models (review)
Steps to Analyze Graph of Polynomial
tan(2a)
=
v
'
1.
2.
3.
4.
• i"\ P
sin
in
- =+
V.2/
—TJ
y-intercepts: f ( 0 )
x-intercept: f(x) = 0
/'crosses / touches axis @ x-inlcrcepts
End behavior: like leading term
5. Find max num turning pts of/'
tan£) = ±
(n-1)
s= - (a + b + c)
or
d(t) =
d = a sin(cal)
cos
where a, b, m constants:
b = damping factor (damping coefficient)
m = mass of oscillating object
\a\ displacement at t = 0
— = period if no damping
I27rl
.«,.._
§10.1 Polar Coordinates
Convert Polar to Rectangular Coordinates
x = r cos 0
y - r sin 0
Convert Rectangular to Polar Coordinates
If x - y = 0 then r = 0, 0 can have any value
else r = Jx2 + y"1
Qi orQiv
§8.1 Inverse Sin, Cos, Tan Fens
Restrict range to [-Ji/2, n!2]
Restrict range to [0,;r]
Restrict range to f— -, -J
§8.2 Inverse Trig Fens (con't)
y = sec"1 x
where |x| > 1 and 0 < y < TI,
where Ixl
< -,
!CSC
X
i- i >
~ 1 and —2; -< Jy —
2'
x - 0, y > 0
x = Q, y < 0
where -co < x < oo and 0 < y < TC
§8.3 Trig Identities
„
s'mB
„
cos 8
tan 9 =coscot 9 =
8
sin 9
csc0 = —
sec#=—
sm9
^
cos d
Pythagorean:
sin" 6 + cos" 0 = 1
tan2 0 + 1 = sec2 0
cot2 0
Heron's Formula
Damped Harmonic Motion
co = frequency (stretch/shrink horizontally)
H < I stretch |tu| > I shrink w < 0 reflect
period = T =\
y = cot x
§9.4 Area of Triangle
K. = - be sin A = - ac sin B = - ab sin C
(J = a cox(wt)
|A| = amplitude (stretch/shrink vertically)
|A| < I shrink |A| > 1 stretch A < 0 reflect
Distance from min to max = 2A
y = sin" (x)
y = cos"' (x)
y = tan"' (x)
a" = b" + c" - 2bc cos A
b2 = a2 + c2 - 2ac cos B
§9.5 Simple & Damped Harmonic Motion
Simple Harmonic Motion
§7.6 Graphing Sinusoiclals
Graphing y = A sin (tox) & y = A cos (cox)
§7.8 Phase Shift = •3
(a
y = A sin (cox - (p) + B
y = A cos (cox - (p) + B
sine
b
c2 = a2 + b2 - 2ab cos C
where degree of nuiner. = n and degree of denom. = in
I . If n < m, horizontal asymptote:.!' - 0 (the x-axis).
2. If n = m, line y = — is a horizontal asymptote.
"m
3. If n = (m + 1 ), quotient from long div is ax + b and line y -- ax
+ b is oblique asymptote.
4. I f n > (m + I ), R has no asymptote,
T-
l+cos a
sinA
sinB
§9.3 Law of Cosines
§5.2 Rational Functions
Finding Horizontal/Oblique Asymptotes of R
I
„ sin a
I l + cos(ff)
§9.2 Law of Sines
6. Behavior near zeros for each x-intercept
7. May need few extra pis to draw fen.
•
COS
tan (3
1 = csc26
§8.4 Sum & Difference Formulae
cos(a ± P) = cos a cos p + sin a sin p
sin(a + P) - sin a cos (3 + cos a sin P
§10.3 Complex Plane & De Moivre's Theorem
Conjugate of
z = x + yi is z = .v + yi
Modulus ofz:
\z\ Vz~f = ^/x2 + y2
Products & Quotients of Complex Nbs (Polar)
z/ = /'/ (cos Oi + i sin O/)
z_? = /•_•> (cos 0? + i sin Or)
ztz2 = rlr2[cos(8l + 62) + i sin(0, + 0 2 )]
-Z 2 =r-2 [cos(0, - 0 2 ) + i sin(0! - 0 2 )]z>^ 0De Moire's Theorem z = r (cos 0 + i sin 0)
zn _ rn[cos ( n0 j + (- sin ( n e)]
Complex Roots n > 2, k = Q. I, 2
§8.5 Double-Angle & Half-Angle Formulae
sin (2o) = 2sina cos a
cos (2a) = cos2 a — sin 2 a
cos (2a) = 1 — 2 sin 2 a = 2 cos2 a — I
nr\B
.
2kn\
• •
zk = VnL cos (Vn H—
TI / + l si
where /t = 0, /, 2
(n - I)
„>/
(n - 1))