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