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Angles (1.1)
Similar Triangles (1.2)
Def’n An angle is formed by rotating a ray
around its vertex.
Def’n Two angles are vertical if their sides form
two pairs of opposite rays.
Def’n A degree () is the measure of an angle
that makes 1/360th of a complete
counterclockwise rotation.
Def’n A transversal is a line that intersects two
parallel lines.
Def’n An angle is acute if it measures between
0 and 90, right if it measures 90, obtuse
if it measures between 90 and 180, and
straight if it measures 180.
Def’n Two angles are complementary if their
measures add to 90 and supplementary
if their measures add to 180.
Def’n An angle is in standard position if its
vertex is the origin and its initial side is
the positive x-axis.
Def’n Two angles are coterminal if their
measures differ by a multiple of 360.
Rule The following pairs of angles are congruent:
(1) vertical angles
(2) corresponding angles
(3) alternate interior angles
(4) alternate exterior angles
Rule The sum of the measures of the angles in
a triangle is 180.
Def’n Two triangles are similar if they have the
same shape.
Def’n Two triangles are congruent if they have
the same size and shape.
Rule Corresponding angles of similar triangles
are congruent.
Rule Corresponding sides of similar triangles
are proportional.
Trig Functions of Angles (1.3)
Even/Odd Identities ()
Rule The trigonometric functions are defined
for an angle  in standard position with
terminal side passing through a point
x, y as follows:
sin  
y
r
cos  
x
r
tan  
y
x
csc  
r
y
sec  
r
x
cot  
x
y
where r 
Reciprocal Identities (1.4)
1
 sin 
csc 
1
 sec 
cos 
1
 cos 
sec 
Quotient Identities (1.4)
sin 
 tan 
cos 
cos 
 cot 
sin 
Pythagorean Identities (1.4)
sin 2   cos 2   1
1  tan 2   sec 2 
1  cot 2   csc 2 
sec    sec 
sin     sin 
csc     csc 
tan     tan 
cot     cot 
Signs of Trigonometric Functions (1.4)
Rule The signs of the trigonometric functions in
The four quadrants are given as follows:
x2  y2 .
1
 csc 
sin 
cos    cos 
1
 cot 
tan 
1
 tan 
cot 
Q

I
0  90 

II
90   180 
III
IV
tan
cot
sec
csc











180   270 






270   360 






sin cos
Trig Functions in Right Triangles (2.1)
Function Values of Special Angles (2.1, 2.2)
Rule The trigonometric functions are defined as
ratios on a right triangle as follows:
Rule Values of the trigonometric functions for
special angles are given as follows:
sin A 
opp
hyp
csc A 
hyp
opp
cos A 
sec A 
adj
hyp
tan A 
hyp
adj
cot A 
opp
adj
adj
opp
Cofunction Identities (2.1)

tan90
sec90


 A   cot A
 A   csc A
sin 90  A  cos A



cot90
csc 90


 A   tan A
 A   sec A
cos 90  A  sin A


Solving Right Triangles (2.4)
Rule Use the Pythagorean Theorem and trig
functions with right triangle definitions to
solve for unknown sides.
Rule Use complementary angles and inverse trig
functions with right triangle definitions to
solve for unknown angles.
Applications of Right Triangles (2.5)
Def’n Bearing is the direction of motion or relative
position, expressed either as (1) a clockwise
angle from due north or (2) an east-west
acute angle from a north-south line.

sin
cos 
tan
cot 
sec
csc
0
0
1
0
U
1
U
30 
12
3 2
3 3
3
2 2
1
1
2
12
3
3 3
2
0
U
0
U
45 
2 2
60 
3 2
90 
1
2
3
2
2
3 2
1 2
 3
 3 3
2
135 
2 2
 2 2
1
1
 2
150 
12
 3 2
 3 3
 3
180 
0
1
0
U
210 
1 2
 3 2
3 3
3
225 
 2 2
 2 2
1
1
240 
 3 2
1 2
3
3 3
2
270 
1
0
U
0
U
300 
 3 2
12
 3
 3 3
2
315 
 2 2
2 2
1
1
2
330 
1 2
3 2
 3 3
 3
360 
0
0
U
1
2
3
1
2
3
 2
2
3
2
2
U
2
 2
2
3
1
2
3
1
3
1
120 
2
2
3
 2
2
U
Radian Measure (3.1)
The Unit Circle (3.3)
Def’n An arc is a portion of a circle intercepted
by a central angle .
Def’n A radian (rad) is the measure of an arc
whose length equals the radius of the circle.
Rule Pi ( ) radians is equal to 180.
Arc Length and Sector Area (3.2)
Def’n The arc length s intercepted by a
central angle with measure  on a
circle of radius r is given by: s  r .
Def’n The area of a sector A of a circle with
radius r swept out by a central angle
1
2
with measure  is given by: Α 
r 2 .
Trig Functions on the Unit Circle (3.3)
Linear Speed and Angular Speed (3.4)
Def’n The unit circle is centered at 0, 0 with
radius 1, and is given by the equation
x 2  y2  1.
Def’n The linear speed v of a point P on a
circle of radius r moving a distance of
s
s in time t is given by: v  .
t
Rule The trigonometric functions are defined
for an arc of length s on the unit circle
with initial point 1, 0 and terminal point
x, y as follows:
csc s 
1
y
sin s  y
1
x
cot s 
x
y
cos s  x
tan s 
y
x
sec s 
Def’n The angular speed  of a point P on
a circle of radius r moving through an
 v
angle  in time t is given by:    .
t r
Graphs of Sine and Cosine (4.1)
Graphs of Secant and Cosecant (4.4)
Def’n A function is periodic if f (t  p )  f (t )
for all t. The number p is the period.
Rule The graphs of f (t )  sec t and
f (t )  csc t both have a period of 2.
Def’n The amplitude of a periodic function
is half the difference between the
maximum and minimum outputs.
Rule The graphs of f (t )  a sec[b(t  d )]  c
and f (t )  a csc[b(t  d )]  c both have
the following features:
Rule The graphs of f (t )  sin t and f (t )  cos t
both have an amplitude of 1 and a
period of 2.
Rule The graphs of f (t )  a sin bt and
f (t )  a cos bt both have an amplitude
(1) vertical stretch of a
(2) period of 2 b
(3) vertical shift of c
(4) phase shift of d
of a and a period of 2 b .
Graphs of Tangent and Cotangent (4.3)
Translations of Sine and Cosine (4.2)
Rule The graphs of f (t )  tan t and
f (t )  cot t both have a period of .
Rule The graphs of f (t )  sin t  c and
f (t )  cos t  c both have a vertical
shift of c.
Rule The graphs of f (t )  sin(t  d ) and
f (t )  cos(t  d ) both have a phase
shift of d.
Rule The graphs of f (t )  a sin[b(t  d )]  c
and f (t )  a cos[b(t  d )]  c both have
the following features:
(1) amplitude of a
(2) period of 2 b
(3) vertical shift of c
(4) phase shift of d
Rule The graphs of f (t )  a tan[b(t  d )]  c
and f (t )  a cot[b(t  d )]  c both have
the following features:
(1) vertical stretch of a
(2) period of  b
(3) vertical shift of c
(4) phase shift of d
Law of Sines (7.1, 7.2)
Complex Numbers in Trigonometric Form (8.2)
Rule The Law of Sines for ABC is given by:
Def’n The trigonometric (or polar) form of the
complex number z  x  yi is given by:
sin A sin B sin C


a
b
c
z  r (cos   i sin  ) , where r  x 2  y 2
and tan   y x .
and is used to solve the SAA, ASA, and SSA cases.
Area of a Triangle (7.1)
Rule The area of ABC is given by:
Area 
1
2
bc sin A 
1
2
ac sin B 
1
2
ab sin C
Law of Cosines (7.3)
Rule The Law of Cosines for ABC is given by:
a 2  b 2  c 2  2bc cos A
2
2
2
b  a  c  2ac cos B
c 2  a 2  b 2  2ab cos C
and is used to solve the SAS and SSS cases.
Area of a Triangle (7.3)
Rule The area of ABC is given by:
Area  s(s  a )(s  b )(s  c )
where s 
1
2
(a  b  c ) is the semiperimeter.
Def’n The rectangular (or standard) form of the
complex number z  r (cos   i sin  ) is given
by: z  x  yi , where x  r cos  and y  r sin  .
Complex Products and Quotients (8.3)
Rule If z1  r1(cos 1  i sin1) and
z 2  r2(cos 2  i sin 2 ) , then
z1z 2  r1r2 cos(1  2 )  i sin(1  2 )
and
z1 r1
 cos(1   2 )  i sin(1   2 ) .
z 2 r2
Complex Powers and Roots (8.4)
Rule If z  r (cos   i sin  ) , then
z n  r n (cos n  i sin n ) .
Rule If z  r (cos   i sin  ) , then the nth
roots of z are given by:
  2k
  2k 

z  n r  cos
 i sin

n
n


for k  0,1,2,..., n  1.
n