Transcript Slide 1
INTRODUCTION TO ENGINEERING TECHNOLOGY
SEVENTH EDITION ROBERT J. POND & JEFFERY L. RANKINEN CHAPTER 6 RIGHT-TRIANGLE TRIGONOMETRY AND GEOMETRY FOR TECHNOLOGISTS
RIGHT-TRIANGLE RELATIONSHIPS
• 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” a B b c A 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 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 ►
c
2
a
2
b
2 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 FUNCTIONS
“sohcahtoa” sin opposite side hyponenuse = a _ c B cos adjacent side hyponenuse c a tan opposite side adjacent side A b 90 o 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 and cotangent.
-1 They are not the same as reciprocal functions known as the secant, cosecant,
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) = 30 o • 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 EXAMPLE
• FIND THE TOTAL IMPEDANCE, Z T , AND THE ANGLE, Θ, OF THE AC CIRCUIT BELOW.
Tan A = opp = -3/4= - 0.75
adj Use inv Tan or Tan of – 36.87
o -1 to find angle measurement To find Z Pythagorean Theorem a 2 + b 2 T , Use the = c 2 2 4 2 + (-3) = c 2 16 + 9 = 25 √25 = c 2 5 = c = Z T