Download Trig Pre-Calc H CC Formulas for Second Sem Final

Trigonometry Second Semester Final Study Sheet
*to convert degrees to radians; multiply the degree by
 rad
* to convert radians to degrees; multiply the radians by
180
180
 rad
Coterminal angles: two angles with the same initial sides but possibly different rotations.
- every angle has infinitely many coterminal angles
-


is coterminal with angles
is coterminal with angles
  360 k (k is an integer)
  2 k
*Length of a Circular Arc
s  r
(  in radians)
Definitions of the Trig functions in Terms of a Unit Circle
sin t = y
csc t = 1
cos t = x
sec t =
1
x
y
x
cot t =
x
y
tan t =
Even Functions
cos(-t) = cos t
sec(-t) = sec t
y
Odd Functions
sin(-t) = -sin t
csc(-t) = -csc t
tan(-t) = -tan t
cot(-t) = -cot t
Reciprocal Identities
1
csc t
1
cos t 
sec t
1
tan t 
cot t
sin t 
1
sin t
1
sec t 
cos t
1
cot t 
tan t
csc t 
Quotient Identities
tan t 
sin t
cos t
cot t 
cos t
sin t
Pythagorean Identities
sin 2 t  cos 2 t  1
1  tan 2 t  sec 2 t
1  cot 2 t  csc 2 t
SOHCAHTOA
Cofunction Identities

The value of a trigonometric function of
is equal to the cofunction of the complement of
Cofunctions of complementary angles are equal.
sin   cos  90   
cos   sin  90   
tan   cot  90   
cot   tan  90   
sec   csc  90   
If

is in radians, replace
90 with
csc   sec  90   

2
.
Definitions of Trigonometric Functions of Any Angle
Let

.
be any angle in standard position and let P = (x, y) be a point on the terminal side of
 . If
r  x 2  y 2 is the distance from (0,0) to (x,y), as shown in the figure above, the six trigonometric
functions of

are defined by the following ratios:
sin  
y
r
cos  
x
r
tan  
y
,x 0
x
r
,y0
y
r
sec   , x  0
x
x
cot   , y  0
y
csc  
Sum and Difference Formulas for Cosines and Sines
cos      cos  cos   sin  sin 
cos      cos  cos   sin  sin 
sin      sin  cos   cos  sin 
sin      sin  cos   cos  sin 
Sum and Difference Formulas for Tangents
tan     
tan   tan 
1  tan  tan 
tan     
tan   tan 
1  tan  tan 
Three forms of the Double-Angle Formula of cos 2
Double Angle Formulas
sin 2  2sin  cos
cos 2  cos 2   sin 2 
cos 2  cos 2   sin 2 
cos 2  2 cos 2   1
tan 2 
cos 2  1  2sin 2 
2 tan 
1  tan 2 
Power Reducing Formulas
1  cos 2
2
1  cos 2
cos 2  
2
1  cos 2
tan 2  
1  cos 2
sin 2  
Half-Angle Formulas
1  cos 
2
2

1  cos 
cos  
2
2

1  cos 
tan  
2
1  cos 
sin


Half-Angle Formulas for Tangent
tan

2

1  cos 
sin 
tan

2

sin 
1  cos 
Product-to-Sum Formulas
1
cos      cos     
2
1
cos  cos   cos      cos     
2
1
sin  cos   sin      sin     
2
1
cos  sin   sin      sin     
2
sin  sin  
Sum-to-product Formulas
sin   sin   2sin
 
cos
 
2
2
 
 
sin   sin   2sin
cos
2
2
 
 
cos   cos   2 cos
cos
2
2
 
 
cos   cos   2sin
sin
2
2
Sum-to-product Formulas
sin   sin   2sin
 
cos
 
2
2
 
 
sin   sin   2sin
cos
2
2
 
 
cos   cos   2 cos
cos
2
2
 
 
cos   cos   2sin
sin
2
2
The Law of Sines
If A,B and C are the measures of the angles of a triangle, and a, b, and c are the lengths of the sides
opposite these angles, then
a
b
c


sin A sin B sin C
The Law of Cosines
If A, B and C are the measures of the angles of a triangle, and a, b and c are the lengths of the sides
opposite these angles, then
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C
The Sign of r and a Point’s Location in Polar Coordinates
The point P = (r,  ) is located r units from the pole. If r>0, the point lies on the terminal side of
.
If r<0, the point lies along the ray opposite the terminal side of
r = 0, the point lies at the pole, regardless of the value of
.
.
If
Relations between Polar and Rectangular Coordinates
x  r cos 
y  r sin 
x2  y 2  r 2
y
tan  
x
Converting a Ponit from Rectangular to Polar Coordinates ( r
1)
Plot the point (x, y).
 0 and 0    2 )
2)
Find r by computing the distance from the origin to (x, y):
3)
Find

using
tan  
y
x
with the terminal side of
The Absolute Value of a Complex Number
The absolute value of the complex number a + bi is
z  a  bi  a2  b2

r  x2  y 2
passing through (x, y).
Polar Form of a Complex Number
The complex number z = a + bi is written in polar form as
z  r (cos   i sin  ) ,
where
a  r cos , b  r sin  , r  a2  b2
and tan  
b
. The value of r is called the
a
modulus (plural: moduli) of the complex number z and the angle
complex number z with 0    2 .

is called the argument of the
Product of Two Complex Numbers in Polar Form
If
z1  r1 (cos 1  i sin 1 ) and z2  r2 (cos 2  i sin 2 )
z1 z2  r1r2 [cos(1  2 )  i sin(1  2 )]
Quotient of Two Complex Numbers in Polar Form
If
z1  r1 (cos 1  i sin 1 ) and z2  r2 (cos2  i sin 2 )
z1 r1
 cos 1   2   i sin 1   2  
z2 r2 
DeMoivre’s Theorem

Let z  r cos 
the nth power, is
 i sin  
be a complex number in polar form. If n is a positive integer, then z to
z n   r  cos   i sin    r n  cos n  i sin n 
n
DeMoivre’s Theorem for Finding Complex Roots


Let w  r cos   i sin  be a complex number in polar form. If
complex nth roots given by the formula
    2 k 
   2 k  
zk  n r cos 
(radians)
  i sin 

n 
n  

 
    360 k 
   360 k  
n
zk  r cos 
 
n
  i sin 


n
The magnitude of v = ai + bj is given by
v  a 2  b2
  (degrees)

w  0 , w has n distinct
Representing Vectors in Rectangular Coordinates
Vector v with initial point
vector
P1  ( x1 , y1 ) and terminal point P2  ( x2 , y2 )
v   x2  x1  i   y2  y1  j
is equal to the position