Outline for Teaching Trigonometry



Trigonometry Outline (Precalculus: Stewart, Redlin, Watson, 3rd edition, 1998)

Roots in Geometry

Definition of Triangle Congruence

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Definition of Triangle Similarity

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Legal Ways to rewrite proportions

Starting Proportion: [pic]

Alternation: [pic]

Inversion: [pic][pic]

Addition: [pic]

Subtraction: [pic]

Any combinations of the above any number of times

Result of similarity: corresponding sides are proportional

This is what makes trigonometry possible

This is why the sine of 30 degrees is always .5

Standard Angle formation

Origin is center of a circle with central angles as our angles

Initial side always on positive x-axis

Terminal side is x degrees in counter-clockwise motion from initial side

Positive and negative angles

Greek letters used for angles

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Angle measure in radians

Why? angle measure tied to size(radius) of circle.

What is a radian?

How many radians are there in 360 degrees?

Radians in terms of pi

Conversion from radians to degrees

Conversion from degrees to radians

Mapping out the circle in radians

Assignment: 6.1

Definition of 3 basic trig functions

Sine

Cosine

Tangent

SOHCAHTOA

Special angles

degrees (PI/6)

degrees (PI/4)

degrees (PI/3)

Quadrantal angles (0,90,180,270,360, ...) [0,PI/2, PI,3PI/2,2PI, ...]

Two special triangles

30-60-90 right triangle

geometry theorem: the side opposite the 30 degree angle in a right triangle is one half the hypotenuse

right triangle (Isosceles Right Triangle)

Use above theorems and the Pythagorean Theorem to develop ratios for all sides

assignment 6.2

Unit Circle (circle whose radius is 1)

"Behavior" Charts

line segments which represent the sine, cosine and tangent

coordinates of point on unit circle (x,y) are (cosine,sine)

signs at quadrantal angles and in each quadrant

incr/decr in each quadrant

trig chart including 0,30,45,60,90,120,135,150,180,210,225,240,270,

negative angles

angles over 360 degrees

assignment 6.3

using a calculator

radians/degrees

for most problems, I require exact answers

using interpolation with tables

New trig functions defined

cosecant

secant

cotangent

representation by line segments

extended trig chart

assignment 6.3

Pythagorean Identities (3)

get from unit circle and segment representations

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demonstration with special angle

demonstrate with calculator

do x,y,r proof

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demonstration with special angle

demonstrate with calculator

do x,y,r proof

[pic]

demonstration with special angle

demonstrate with calculator

do x,y,r proof

derivative identity forms from the Pythagorean Identities

do all functions in terms of sine,cosine,tangent,cosecant,secant,cotangent

assignment 7.1

Reciprocal Identities (Review) (3)

1/sine = cosecant

1/cosine = secant

1/tangent = cotangent

Ratio Identities (2)

Tangent = sine/cosine

cotangent = cosine/sine

More Identities

Sum and Difference (6)

Double Angle, Half Angle (6)

assignments 7.1, 7.2, 7.3

Graphs of Sine and Cosine functions

amplitude

period

shift

sum curves

trig graph +- constant

trig graph +- linear graph

3. trig graph +- trig

Other Trig Graphs

Tangent

Cosecant

Secant

Cotangent

assignment 5.3, 5.4

Law of Sines, Cosines

Assignment 6.4, 6.5

Trigonometric Equations

Assignment 7.5

Trigonometric Forms of Complex Numbers

DeMoivre’s Theorem

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