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Lesson 4.3: Equivalent Trigonometric Expressions Objectives: Using Equivalent Trigonometric Expressions to Evaluate Primary / Reciprocal Trigonometric Expressions State the co-function formulas (Trigonometric Identities Featuring П/2) Sin x = Cos (П/2 – x) Tan x = Cot (П/2 – x) Csc x = Sec (П/2 – x) Cos x = Sin (П/2 – x) Cot x = Tan (П/2 – x) Sec x = Csc (П/2 – x) Sin (П/2 + x) = cos x Tan (П/2 + x) = - cot x Csc (П/2 + x) = sec x Cos (П/2 + x) = -sin x Cot (П/2 + x) = - tan x Sec (П/2 + x) = - csc x Example: Given that sin П/5 = 0.5878, use equivalent trigonometric expressions to evaluate the following, to four decimal places. a) cos 3П/10 Solution: Since an angle of 3П/10 lies in the first quadrant it can be expressed as the different between П/2 and an angle a. Let find the measure of angle a. 3П/10 = П/2 – a Now apply the cofunction identity: a = П/2 - 3П10 3П/10 = П/2 - a = 5П/10 - 3П/10 = 2П10 = П/5 cos (3П/10) = cos (П/2 –a) = sin a = sin П5 = 0.5878 b) cos 7П/10 Solution: PRACTICE 4.3: Name ____________________ Date _____________________ Knowledge Write each of the following in terms of the cofunction identity: 1. sin 12 2. sin 5 18 6. cos 5. cos 2 5 9 3. sin 5 8 4. sin 5 12 7. cos 7 36 8. cos 2 9 Application Fill in the blanks with the appropriate function name: 2 _________ 3 6 11 19 sin 10. _____________ 60 60 7 1 11. cos 18 _________ 9 9. sin B For right triangle ABC: 12. If sin A of cos B? 3 , what is the value 3 13. If cos A=0.109, what is sin A ? 2 11 79 0.9816 , what is sin 14. If cos ? 180 180 c a A C b P’(b,a) P(a,b) Thinking 15. The reason for the cofunction relationships can be seen from the diagram. If the sum of the measures of POA and P ' OA θ O , then P and P’ are symmetric with respect to the line y = x. 2 Also, if P=(a,b), then P’=(b,a) and sin y-coordinate of P = x-coordinate of P’ = cos . Use this information to derive similar cofunction relationships for 2 is tangent and cotangent, as well as secant and cosecant. A 3.3.2 HOME ACTIVITY: Equivalent Trigonometric Expressions (Answers) Knowledge Write each of the following in terms of the cofunction identity: 1. sin 5 12 2. sin 5 2 sin 18 9 6. cos 12 5. cos cos 2 cos 5 10 9 sin 7 18 3. sin 5 cos 8 8 4. sin 5 cos 12 12 7. cos 7 11 sin 36 36 8. cos 2 5 sin 9 18 Application Fill in the blanks with the appropriate function name: 2 cos 3 6 11 19 sin 10. cos 60 60 7 1 11. cos 18 csc 9 9. sin For right triangle ABC: 12. 3 3 13. 0.109 14. 0.9816 Thinking tan cot 2 15. cot tan 2 sec csc 2 csc sec 2