Trig Ratios in Right Triangles

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Advanced Mathematical Concepts Chapter 5 Lesson 5-2 Example 1 Find the values of the sine, cosine, and tangent for C. First, find the length of AB . (AB)2 + (AC)2 = (BC)2 Pythagorean Theorem (AB)2 + 302 = 452 Substitute 30 for AC and 45 for BC. (AB)2 = 1125 AB = 1125 or 15 5 Take the square root of each side. Disregard the negative root. Then write each trigonometric ratio. side opposite side adjacent side opposite sin C = cos C = tan C = hypotenuse hypotenuse side adjacent 15 5 5 30 2 15 5 5 sin C = or cos C = or tan C = or 45 3 45 3 30 2 Trigonometric ratios are often simplified, but never written as mixed numbers. Example 2 PHYSICS When an object rests on a frictionless surface, the forces acting upon it are gravity and the normal force. The normal force FN is always perpendicular to the surface upon which the object sits, and is defined by FN = mg cos , where m is the mass of the object, g is acceleration due to gravity, and is the angle of inclination of the surface. Suppose a 30 kilogram object is resting on a surface that is at a 25° incline. What is the normal force, in Newtons, acting upon the object given that acceleration due to gravity is 9.8 m/s2? FN = mg cos Definition of normal force FN = (30)(9.8) cos 25° Substitute 30 for m, 9.8 for g, and 25° for . FN (30)(9.8)(0.906307787) Use a calculator to find the cosine ratio. FN 266.4544894 Use a calculator to find the product. The normal force acting on the object is about 266.4545 Newtons. Advanced Mathematical Concepts Chapter 5 Example 3 2 b. If sec = 3.05, find cos . a. If tan = , find cot . 5 1 cos = 1 sec cot = tan 1 cos = or about 0.3279 1 5 3.05 cot = or 2 2 5 Example 4 Find the values of the six trigonometric ratios for R. First determine the length of the hypotenuse. (RS)2 + (ST)2 = (RT)2 Pythagorean Theorem 144 + 196 = (RT)2 Substitute 12 for RS and 14 for ST. 340 = (RT)2 340 or 2 85 = RT Disregard the negative root. side opposite side adjacent side opposite sin R = cos R = tan R = hypotenuse hypotenuse side adjacent 14 7 85 12 6 85 14 7 sin R = or cos R = or tan R = or 2 85 85 2 85 85 12 6 hypotenuse hypotenuse side adjacent csc R = sec R = cot R = side opposite side adacent side opposite 2 85 85 2 85 85 12 6 csc R = or sec R = or cot R = or 14 7 12 6 14 7 .

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