Overview & Fundamentals of Trigonometry
Branch of Geometry: http://en.wikipedia.org/wiki/Trigonometry
Definition: Trigonometry is the study of the relationships
between specific sides & angles of triangles. (Right ∆)
http://mathworld.wolfram.com/Trigonometry.html
Y
Hypotenuse
Opposite
X
Adjacent
Z
Relations of Angle X between Opposite, Adjacent, and Hypotenuse sides.
Relations for Angle Y can be defined by analogous ratios. Can you define Y?
Relationships: Sin X = O/H Cos X = A/H Tan X = O/A
Defined as Inverses: Csc X = 1/Sin X Sec X = 1/Cos X Cot V = 1/Tan X
The Fundamental Geometry is Euclidean based on a Flat Plane…
http://en.wikipedia.org/wiki/Euclidean_geometry
Trigonometric Functions (circular functions) are defined by unit circle:
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Euclidean Theorem: <A + <B + <C = 1800
Pythagorean Theorem: a2 + b2 = c2 Î sin2Ө + cos2Ө = 1
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References: http://en.wikipedia.org/wiki/Trigonometry & http://mathworld.wolfram.com/Trigonometry.html
Directed Angles & Co-Terminal Angles & General Angles
Directed Angles are CW = ( + ) or CCW = ( − )
Co-Terminal ( + & − ) Angles have |AV| sum equal to 3600
General (Big) Angles are greater than 3600
Directed, Co-Terminal, and General angles are vital for future work.
Students need to practice with these relationships until they are innate.
Given: +1400
Determine & Locate Co-Terminal: ____
Given: −2300
Determine & Locate Co-Terminal: ____
Directed (ASTC) or Reference (Trigonometric) Triangles
Trigonometry allows for angles over 1800 called Directed or Reference Triangles
Sin A = O/ H
Csc A = 1/Sin
Cos A = A/H Tan A = O/A
Sec A = 1/Cos
Cot A = 1/Tan
Csc A = H/O Sec A = H/A Cot A = A/O
To determine ASTC values use the angle at the origin within the triangles.
Triangles & (+) (−) signs show for all Trigonometric (+) (−) Values in Quads: I, II, III, IV
Quadrants
Sin
Cos
Tan
I
+/+
+/+
+/+
II
+/+
−/+
−/+
III
−/+
−/+
−/− = +
IV
−/+
+/+
−/+
Inverse
1/Sin
1/Cos
1/Tan
Inverse
Csc
Sec
Cot
Reference ( I, II, III, IV ) values are very much needed for future Math.
Students need to review with these relationships until they are innate.
Extra Special Trigonometric Angles: 00 900 1800 2700 3600+
The Triangles represented below have Special Angles that “do not exist”.
All small sides of these Triangles = 0 while the hypotenuse & adjacent side = 4.
Therefore this is an impossibility thus the Triangles do not actually exist.
Image below provides a illustration for values of non-existing Special Angles.
II
I
900
4
4
4
4
4
4
00
1800
4
4
III
IV
2700
0/4 = 0
Quadrants
Sin
Cos
Tan
I
4/0 = ∞
4/4 = 1
0
II
1
1
0
0
∞
III
0
1
IV
1
0
0
∞
1/Sin
1/Cos
1/Tan
Students have difficulity with Extra Special Angles, thus illustration helps!
Trigonometry uses unique angles of 30, 45, 60 degrees in special situations.
These angles are special in that some have irrational values.
The Equilateral Triangle & Square allow for easy determination of these
unique Co-Function (Irrational) Relations for Trigonometric Functions.
Co-Function Relation can be viewed as a Complementary ( 900 ) Relation.
Cos A + Sec A = 900
Tan A + Cot A = 900
Sin A + Csc A = 900
Co-Function Relation ( Rational & Irrational ) Table of Values
Sin
300
1/2
450
1/√
600
/
Cos
/
1/√
1/2
Tan
1/√
1
√
Cot
√
1
1/√
Sec
2/√
√
2
Csc
2
√
2/√
Basic (6) Trigonometry Curves & Degree / Radian Measure Relationships
A Degree measure equals 1/360 of a Circle since a Circle contains 360 Degrees.
A Radian measure equals the ratio (180/Pi): approximately 57.2958 degrees.
Pi (
) is a Constant which equals the ratio (C/D): Circle Circumference to the Radius.
Basic Relationships: Pi = 1800 2Pi = 3600
C=2 Pi x R or C = Pi x D
The Six Basic Curves of Trigonometry represent movement of a Right Triangle about a Circle.
Sin A = O/H
Csc A = H/O or 1/Sin A
Cos A = A/H
Sec A = H/A or 1/Cos A
Tan A = O/A
Cot A = A/O or 1/Tan A
@ Students need to after much practice sketch from memory basic trigonometric curves! @
Note: Curves on next page with all Trigonometric Functions from - 2 Π to + 2 Π or for (2) cycles.
Sin A: - 2 Π to + 2 Π (Note: Bumps of Sin = U’s of Csc) Csc A:
Cos A: - 2 Π to + 2 Π (Note: Bumps of Cos = U’s of Sec)
- 2 Π to + 2 Π
Sec A: - 2 Π to + 2 Π
Tan A: - 2 Π to + 2 Π (Note: Similarities&Differences) Cot A: - 2 Π to + 2 Π
Images from Free Online Graphing Calculator (GraphCalc) http://www.graphcalc.com/
Reference: Dolciani, Mary P., Modern Algebra & Trigonometry, Houghton Mifflin Co., New York, NY
Reference: Dolciani, Mary P., Modern Introductory Analysis, Houghton Mifflin Co., New York, NY
Thomas E. Love *** Malone University *** 2012/2013