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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” B a b Sides are named with lower case letters; such as a, b, c Think of a as altitude and b as base. c is the c hypotenuse. Angles are identified by capital letters and are A 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 a b 2 2 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” opposite side sin hyponenuse adjacent side cos hyponenuse opposite side tan adjacent side = a_ c = B c b_ c = a_ b a A 90o b 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. 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 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 Theorem a2 + b2 = c2 42 + (-3) 2 = c2 16 + 9 = 25 √25 = c2 5 = c = ZT