INTRODUCTION TO INTRODUCTION TO ENGINEERING ENGINEERING TECHNOLOGYTECHNOLOGYSEVENTH EDITIONSEVENTH EDITIONROBERT J. POND & JEFFERY L. ROBERT J. POND & JEFFERY L. RANKINENRANKINEN

CHAPTER 6

RIGHT-TRIANGLE TRIGONOMETRY AND GEOMETRY FOR TECHNOLOGISTS

RIGHT-TRIANGLE RIGHT-TRIANGLE RELATIONSHIPSRELATIONSHIPS• THE 3-4-5 TRIANGLE HAS LEGS MEASURING 3 AND 4

UNITS AND THE HYPOTENUSE MEASURING 5 UNITS

• IT IS A RIGHT TRIANGLE

• SOMETIMES CALLED THE “MAGIC 3-4-5 TRIANGLE”

B

A

a

b

c

Sides are named with lower case letters; such as a, b, c Think of a as altitude and b as base. c is the hypotenuse. Angles are identified by capital letters and are opposite the sides with the same letter.

RIGHT-TRIANGLE RELATIONSHIPSRIGHT-TRIANGLE RELATIONSHIPS• Carpenters still use 3-4-5 Triangle

•When the sides of a right triangle are all integers, it is called a Pythagorean Triple

• PYTHAGOREAN THEOREM

222 bac ► Three angles always total

180º► For a right triangle, angles

A and B total 90º► 2 angles that add to 90º

are called complementary

TRIGONOMETRIC FUNCTIONSTRIGONOMETRIC FUNCTIONS

sideadjacent

side oppositetan

hyponenuse

sideadjacent cos

hyponenuse

side oppositesin

a _ c

=

= b_ c

= a _ b

c

a

b

90o

Inverse Trig functions are used to find angle measurements when you know the sin, cos, or tan. They are written as, for example, inv sin or sin-1 They are not the same as reciprocal functions known as the secant, cosecant, and cotangent.

“sohcahtoa”

A

B

OTHER TRIGONOMETRIC FUNCTIONS

• INVERSE FUNCTIONS, SUCH AS TAN-1, SIN-1 , AND COS -1 , WILL GIVE THE ANGLE VALUE, WHEN THE FUNCTION VALUE IS KNOW. THESE ARE ALSO CALLED ARC TANGENT, ARC SINE, AND ARC COSINE.

• THE COS-1 (0.500) = 30o

• THE COSECANT, SECANT, AND COTANGENT FUNCTIONS ARE RECIPROCALS OF THE SINE, COSINE AND TANGENT FUNCTIONS• HYPOTENUSE OPPOSITE SIDE

HYPOTENUSE ADJACENT SIDE

ADJACENT SIDE

OPPOSITE SIDE

=

=

=

TRIG EXAMPLETRIG EXAMPLE• FIND THE TOTAL IMPEDANCE, ZT, AND THE ANGLE, Θ, OF THE AC CIRCUIT BELOW.

Tan A = opp = -3/4= - 0.75

adj Use inv Tan or Tan-1 to find angle measurement of – 36.87o

To find ZT , Use the Pythagorean Theorema2 + b2 = c2

42 + (-3) 2 = c2

16 + 9 = 25√25 = c2

5 = c = ZT