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Introduction to Trigonometry
What is Trigonometry?
• Trigonometry is the study of how the sides and angles of a triangle are related to each other.
• It's all about triangles!
A
Right Triangle
Opposite
Hypotenuse
Adjacent
Same Right Triangle – Different Angle
Hypotenuse
B
Adjacent
Opposite
Trig Definitions:
• Sine = opposite/hypotenuse • Cosine = adjacent/hypotenuse • Tangent = opposite/adjacent• Cosecant = hypotenuse/opposite• Secant = hypotenuse/adjacent• Cotangent = adjacent/opposite
A Way To Remember
• Sin = Opposite/Hypotenuse Oprah Had
• Cos = Adjacent/Hypotenuse A Huge
• Tan = Opposite/Adjacent Old Afro
y
x
y
x
r
O
x,y
r = x + y2 2 2
Definitions of Trig Functions
• Sin = y / r• Cos = x / r• Tan = y / x• Csc = r / y• Sec = r / x• Cot = x / y
O
O
O
OOO
Radius = 1
The Unit Circle
y
x
1/2
30
x,y
r = x + y2 2 2
1
√3/2
y
x
√/2/2
45
x,y
r = x + y2 2 2
1
√2/2
y
x
1/2
60
x,y
r = x + y2 2 2
1
√3/2
Trigonometric Functions on a Rectangular Coordinate System
x
y
Pick a point on theterminal ray and drop a perpendicular to the x-axis.
(The Rectangular Coordinate Model)(The Rectangular Coordinate Model)
Trigonometric Functions on a Rectangular Coordinate System
x
y
Pick a point on theterminal ray and drop a perpendicular to the x-axis.
ry
x
The adjacent side is xThe opposite side is yThe hypotenuse is labeled rThis is called a REFERENCE TRIANGLE.
y
x
x
yx
r
r
x
y
r
r
y
cottan
seccos
cscsin
Trigonometric Values for angles in Quadrants II, III and IV
x
y
Pick a point on theterminal ray and drop a perpendicular to the x-axis.y
x
r
y
x
x
yx
r
r
x
y
r
r
y
cottan
seccos
cscsin
Trigonometric Values for angles in Quadrants II, III and IV
x
yPick a point on theterminal ray and raise a perpendicular to the x-axis.
Trigonometric Values for angles in Quadrants II, III and IV
x
yPick a point on theterminal ray and raise a perpendicular to the x-axis.
y
x
x
yx
r
r
x
y
r
r
y
cottan
seccos
cscsin
x
r y
Important! The is
ALWAYS drawn to the x-axis
Signs of Trigonometric Functions
x
y
AAll are positive in QI
TTan (& cot) are positive in QIII
SSin (& csc) are positive in QII
CCos (& sec) are positive in QIV
Signs of Trigonometric Functions
x
y
AAll
TTake
SStudents
CCalculus
is a good way toremember!
Trigonometric Values for Quadrantal Angles (0º, 90º, 180º and 270º)
x
y
º
Pick a point one unit from the Origin.
(0, 1)
r
x = 0y = 1r = 1
090cotundefined is 90tan
undefined is 90sec090cos
190csc190sin
Trigonometric Ratios may be found by:
45 º
1
1
2Using ratios of special trianglesUsing ratios of special triangles
145cot145tan
245sec2
145cos
245csc2
145sin
For angles other than 45º, 30º, 60º or Quadrantal angles, you will For angles other than 45º, 30º, 60º or Quadrantal angles, you will need to use a calculator. (Set it in Degree Mode for now.)need to use a calculator. (Set it in Degree Mode for now.)
For Reciprocal Ratios, use the facts:For Reciprocal Ratios, use the facts:
tan1cot
cos1sec
sin1csc
AcknowledgementsThis presentation was made possible by
training and equipment provided by an Access to Technology grant from Merced College.
Thank you to Marguerite Smith for the model.
Textbooks consulted were: Trigonometry Fourth Edition by Larson & Hostetler Analytic Trigonometry with Applications Seventh Edition
by Barnett, Ziegler & Byleen