What I Need to Know from Trig
[the stuff I was supposed to have learned last year]
[By the way, you’ll need to be able to do this WITHOUT a calculator!]
Note: AP Calculus AB never uses degrees – only radians
0° = 0
30° =
45° =
60° =
90° =

6

4

3

2
180° =

270° =
3
2
Opposite
Hypotenuse

360° = 2 
Adjacent
In AP Calculus you should memorize the six trig function values above.
Right Triangle
SOHCAHTOA
opp
hyp
1
hyp
csc  =

sin  opp
adj
hyp
1
hyp
sec  =

cos  adj
sin  =
opp
adj
1
adj cos 
cot  =


tan  opp sin 
cos  =
tan  =
(0,1)
Circular Functions
(x,y)
y
r
x
cos  =
r
y
tan  =
x
sin  =
1 radian  57.3°
r
y
r
sec  =
x
x
cot  =
y
csc  =
rr
( -1, 0)
r

(0, -1)

2
3.14
6.28
 1.57radians
  3.14radians
3
 4.71radians
2
2  6.28radians
4.71
The most important trig formula:
sin2 cos 2
from which we can derive other useful trig formulas.
(Divide all parts by sin2andwe get
1 + cot2csc2 
(Divide all parts by cos2  ) and we get
tan2  + 1 = sec2 
y
(1, 0) (1, 0)
x
 1.57
(cos  , sin  )
A - all trig functions positive in Quad I
II
I
S
A
T
C
S - sin & csc functions positive in Quad II
T – tan & cot functions positive in Quad III
III
C – cos & sec function positive in Quad IV
IV
Trig Graphs
y  cos x
y  sin x
y  csc x
y  tan x
y  sec x
Reference Angles (
'
) ---- Positive, acute angles
Quad I -
 ' 
Quad II -
 '   
 '  180 
Quad III -
 '   
 '   180
Quad IV -
 '  2 
 '  360 
y  cot x
reference angle is itself
Special Right Triangles
1
does not need to be rationalized (Hooray!). It is sometimes easier (at least for me) to
2
In the AP- world, the answer
recall trig values using triangles rather than using the unit circle.
45°
60°
2
1
2
30° =
1

45°
3

1
sin 
6 2
csc

6
2
1

3
cos 
6
2

2
sec 
6
3

1
tan 
6
3

cot  3
6
sin


1
2
4

csc  2
4
cos
----
For example: sin-1  =
means the arc (or angle) whose sine is equal to some given value.

1
means  
4
2
INVERSE FUNCTIONS ARE NOT RECIPROCALS!!!

1
2
4

sec  2
4
Inverse trig functions
sin-1  = arcsin 


1
4

cot  1
4
tan
What I Need to Know from Pre Calculus
Linear equations:
slope-intercept form: y  mx  b
point-slope form: y  y1  m  x  x1 
Parallel lines have equal slopes (except vertical lines which have undefined slopes)
Perpendicular lines have slopes whose product is –1 (opposite reciprocals)
y y1  y2
slope = m =

x x1  x2
Distance formulas:
from point  x1 , y1  to  x2 , y2  :
d
 x1  x2 
  y1  y2 
2
from point  x1 , y1  to line Ax  By  C  0 :

Domain and Range:
d
2
Ax1  By1  C
A2  B 2
Interval Notation
1x 1 can be written as [ -1, 1]

1x 1 can be written as (1,1]
1x1 can be written as 1,1 
1x1 can be written as [1,1)
Closed bracket is inclusive, open parenthesis is exclusive.
Always use ( ) whenever is involved.
Consider the following function:
f(x) = x2
Domain: ,
Range: [0,)
Geometry formulas:
1
bh
2
Area
Triangle:
Surface area
Sphere: 4 r 2
Volume
Cone:
Trapezoid:
1
h(b1  b2 )
2
Circle:  r 2
Lateral area of cylinder: 2 rh
1 2
4 3
r h
r
Sphere:
Cylinder:  r 2 h
3
3
Prism: Bh where B is the area of the base
1
Bh where B is the area of the base
Pyramid:
3
Symmetry of functions:
Even functions have the property f ( x)  f ( x) , & the graph of an even function is symmetric with respect to the y-axis
Odd functions have the property f ( x)   f ( x) , & the graph of an odd function is symmetric with respect to the origin.
Zeros of polynomials:
The solutions to ax2  bx  c  0 are x 
b  b 2  4ac
2a
If P ( x ) is a polynomial with leading coefficient a and constant term c, then any rational zeros must be of the form p q
where p is a divisor of c and q is a divisor of a.
Exponents and logarithms:
A.
b x  y  logb y  x
bx b y  bx y
logb x  logb y  logb xy
B.
b
 bx y
by

logb x  logb y  logb 

C.
b 
logb x y  y logb x
D.
If b x  b y or if logb x  logb y , then x  y
E.
logb b x  x and blog x  x
log x  log10 x and ln x  log e x
log c a
logb a 
log c b
F.
G.
x
x
y
 b xy
x

y
b
Transformations of graphs:
A.
B.
C.
D.
E.
F.
y  f ( x  a) is the graph of y  f ( x) shifted horizontally a units (to the right if a  0 and to the left if
a  0)
y  f ( x)  a is the graph of y  f ( x) shifted vertically a units (up if a  0 and down if a  0 )
y  af ( x) is the graph of y  f ( x) stretched or shrunk vertically by a factor of a (stretched if a  1 and
shrunk if 0  a  1 )
y  f (ax) is the graph of y  f ( x) stretched or shrunk horizontally by a factor of a (stretched if 0  a  1
and shrunk if a  1 )
y   f ( x) is the graph of y  f ( x) reflected over the x-axis
y  f ( x) is the graph of y  f ( x) reflected over the y-axis
Sequences and series:
Arithmetic: an  an 1  d ; an  a1  (n  1)d ; Sn 
Geometric: an  an 1 r ; an  a1 r n1 ; Sn 
Special products and factoring
Sum/difference of two cubes:
n  a1  an 
2
a
a1 (1  r )
; S  1 , r  1
1 r
1 r
n

  a  b  a
a3  b3   a  b  a 2  ab  b2
a 3  b3
2
 ab  b2


Basic graphs:
y x
y  ax , a  1
y
1
x
y  ax , 0  a  1
y x
y  log a x, a  1
y  x n , n even
y  x n , n odd
y  log a x, 0  a  1