Outline
for Teaching Trigonometry
Class Notes:
I.
Roots in Geometry
A. Definition
of Triangle Congruence
1. 
2. 
B. Definition
of Triangle Similarity
1. 
2.
3. 
C. Legal
Ways to rewrite proportions
1. Starting
Proportion: 
2. Alternation: 
3. Inversion: 
4. Addition: 
5. Subtraction: 
6. Any
combinations of the above any number of times
7. Result
of similarity: corresponding sides are proportional
8. This
is what makes trigonometry possible
9. This
is why the sine of 30 degrees is always .5
II. Standard
Angle formation
A. Origin
is center of a circle with central angles as our angles
B. Initial
side always on positive x‑axis
C. Terminal
side is x degrees in counter‑clockwise motion from initial side
D. Positive
and negative angles
E. Greek
letters used for angles
III. Angle
measure in radians
A. Why?
angle measure tied to size(radius) of circle.
B. What
is a radian?
C. How
many radians are there in 360 degrees?
D. Radians
in terms of pi
E. Conversion
from radians to degrees
F. Conversion
from degrees to radians
G. Mapping
out the circle in radians
IV. Definition
of 3 basic trig functions
A. Sine
B. Cosine
C. Tangent
D. SOHCAHTOA
V. Special angles
A. degrees
(PI/6)
B. degrees
(PI/4)
C. degrees
(PI/3)
D. Quadrantal
angles (0,90,180,270,360, ...) [0,PI/2, PI,3PI/2,2PI, ...]
VI. Two special triangles
A. 30-60-90
right triangle
1. geometry
theorem: the side opposite the 30 degree angle in a right triangle is one half
the hypotenuse
B. right
triangle (Isosceles Right Triangle)
1. Use
above theorems and the Pythagorean Theorem to develop ratios for all sides
a) assignment
1.1
VII.
Unit Circle (circle whose radius is 1)
A. "Behavior"
Charts
1. line
segments which represent the sine, cosine and tangent
2. coordinates
of point on unit circle (x,y) are (cosine,sine)
3. signs
at quadrantal angles and in each quadrant
4. incr/decr
in each quadrant
5. trig
chart including 0,30,45,60,90,120,135,150,180,210,225,240,270,
6. negative
angles
7. angles
over 360 degrees
a) assignment
1.3
8. using
a calculator
a) radians/degrees
b) for
most problems, I require exact answers
9. using
interpolation with tables
VIII.
New trig functions defined
A. cosecant
B. secant
C. cotangent
D. representation
by line segments
E. extended
trig chart
1. assignment
1.4
IX. Pythagorean Identities (3)
A. get
from unit circle and segment representations
1. 
2. 
3. 
4. demonstrate with calculators that these truly
are identities
B. derivative
identity forms from the Pythagorean Identities
C. do
all functions in terms of sine,cosine,tangent,cosecant,secant,cotangent
X.
Reciprocal Identities (Review) (3)
A. 1/sine
= cosecant
B. 1/cosine
= secant
C. 1/tangent
= cotangent
XI.
Ratio Identities (2)
A. Tangent
= sine/cosine
B. cotangent
= cosine/sine
assignments
1.5, 2.1,2,2, - 3.5
XII. Graphs of Sine and Cosine
functions
A. amplitude
B. period
C. shift
D. sum
curves
1. trig graph +‑ constant
2.
trig graph +‑ linear graph
3.
trig graph +‑ trig
a) assignment 4.1, etc.
XIII. Other Trig Graphs
A. Tangent
B. Cosecant
C. Secant
D. Cotangent
