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Multiple-Angle and Product-to-Sum Formulas MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Objectives In this lesson we will learn to: use multiple-angle formulas to rewrite and evaluate trigonometric functions, use power-reducing formulas to rewrite and evaluate trigonometric functions, use half-angle formulas to rewrite and evaluate trigonometric functions, use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions, use trigonometric formulas to rewrite real-life models. J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Background Recall the sum and difference of angle formulas we have learned previously: sin(u + v ) = sin u cos v + cos u sin v sin(u − v ) = sin u cos v − cos u sin v cos(u + v ) = cos u cos v − sin u sin v cos(u − v ) = cos u cos v + sin u sin v tan u + tan v tan(u + v ) = 1 − tan u tan v tan u − tan v tan(u − v ) = 1 + tan u tan v J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Double-Angle Formulas A double-angle is generally expressed as 2u. Thus the double-angle formulas are: sin 2u = 2 sin u cos u J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Double-Angle Formulas A double-angle is generally expressed as 2u. Thus the double-angle formulas are: sin 2u = 2 sin u cos u cos 2u = cos2 u − sin2 u 2 tan u tan 2u = 1 − tan2 u J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Find the solutions to the following equation in the interval [0, 2π). sin 2x + cos x J. Robert Buchanan = 0 Multiple-Angle and Product-to-Sum Formulas Example Find the solutions to the following equation in the interval [0, 2π). sin 2x + cos x = 0 2 sin x cos x + cos x = 0 cos x(2 sin x + 1) = 0 1 2 π 3π 7π 11π , , , 2 2 6 6 cos x = 0 or sin x = − x J. Robert Buchanan = Multiple-Angle and Product-to-Sum Formulas Example Find the exact values of sin 2u, cos 2u, and tan 2u given that cot u = −5 and 3π/2 < u < 2π. J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Find the exact values of sin 2u, cos 2u, and tan 2u given that cot u = −5 and 3π/2 < u < 2π. Using right triangle trigonometry we see that √ √ − 26 5 26 1 sin u = , cos u = , tan u = − . 26 26 5 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Find the exact values of sin 2u, cos 2u, and tan 2u given that cot u = −5 and 3π/2 < u < 2π. Using right triangle trigonometry we see that √ √ − 26 5 26 1 sin u = , cos u = , tan u = − . 26 26 5 √ 26 sin 2u = 2 sin u cos u = 2 − 26 ! √ ! 5 26 5 =− 26 13 25 1 12 − = 26 26 13 −2/5 2 tan u 5 = =− 1 − 1/25 12 1 − tan2 u cos 2u = cos2 u − sin2 u = tan 2u = J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Verify the following identity. (sin x + cos x)2 = 1 + sin 2x J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Verify the following identity. (sin x + cos x)2 = 1 + sin 2x sin2 x + 2 sin x cos x + cos2 x 2 = 2 (sin x + cos x) + (2 sin x cos x) = 1 + sin 2x J. Robert Buchanan = Multiple-Angle and Product-to-Sum Formulas Power-Reducing Formulas Power-reducing formulas replace powers of trigonometric functions with simpler expressions. Power-Reducing Formulas sin2 u = cos2 u = tan2 u = J. Robert Buchanan 1 − cos 2u 2 1 + cos 2u 2 1 − cos 2u 1 + cos 2u Multiple-Angle and Product-to-Sum Formulas Example Use power-reducing formulas to rewrite the following expression in terms of the first power of cosine. sin2 x cos2 x = J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use power-reducing formulas to rewrite the following expression in terms of the first power of cosine. 1 + cos 2x 1 − cos 2x 2 2 sin x cos x = 2 2 1 = (1 − cos 2x)(1 + cos 2x) 4 1 = (1 − cos2 2x) 4 1 = sin2 2x 4 1 1 − cos 4x = 4 2 1 = (1 − cos 4x) 8 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Half-Angle Formulas Closely related to the power-reducing formulas are the half-angle formulas which are obtained by replacing u by u/2. Half-Angle Formulas u sin 2 u cos 2 u tan 2 r 1 − cos u 2 1 + cos u = ± 2 1 − cos u sin u = = sin u 1 + cos u = ± r u u The signs of sin and cos depend on the quadrant in which 2 2 u falls. 2 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use the half-angle formulas to determine the exact values of sin 165◦ = cos 165◦ = tan 165◦ = J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use the half-angle formulas to determine the exact values of s √ ◦ r ◦ 1 − cos 330 1 − 3/2 330 ◦ sin 165 = sin = = 2 2 2 cos 165◦ = tan 165◦ = J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use the half-angle formulas to determine the exact values of s √ ◦ r ◦ 1 − cos 330 1 − 3/2 330 ◦ sin 165 = sin = = 2 2 2 s r √ ◦ 330 1 + cos 330◦ 1 + 3/2 ◦ cos 165 = cos =− =− 2 2 2 tan 165◦ = J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use the half-angle formulas to determine the exact values of s √ ◦ r ◦ 1 − cos 330 1 − 3/2 330 ◦ sin 165 = sin = = 2 2 2 s r √ ◦ 330 1 + cos 330◦ 1 + 3/2 ◦ cos 165 = cos =− =− 2 2 2 s s √ √ 1 − 3/2 2− 3 √ √ =− tan 165◦ = − 1 + 3/2 2+ 3 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Product-to-Sum Formulas If we combine the sum and difference of angles formulas we can derive the product-to-sum formulas. Product-to-Sum Formulas sin u sin v = cos u cos v = sin u cos v = cos u sin v = 1 [cos(u − v ) − cos(u + v )] 2 1 [cos(u − v ) + cos(u + v )] 2 1 [sin(u + v ) + sin(u − v )] 2 1 [sin(u + v ) − sin(u − v )] 2 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use the product-to-sum formulas to write the following as a sum or difference. sin π π cos 4 12 = J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use the product-to-sum formulas to write the following as a sum or difference. π π π 1 h π π π i sin cos = sin + + sin − 4 12 2 4 12 4 12 π i 1 h π = sin + sin 2 3 6 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use the product-to-sum formulas to write the following as a sum or difference. π π π 1 h π π π i sin cos = sin + + sin − 4 12 2 4 12 4 12 π i 1 h π = sin + sin 2" 3 # 6 √ 1 3 1 + = 2 2 2 √ 3+1 = 4 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Sum-to-Product Formulas Occasionally we may wish to rewrite a sum or difference of trigonometric functions as a product. Sum-to-Product Formulas u−v u+v cos 2 sin 2 2 u+v u−v 2 cos cos 2 2 u−v u+v 2 cos cos 2 2 u+v u−v −2 sin sin 2 2 sin u + sin v = sin u − sin v = cos u + cos v = cos u − cos v = J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use the sum-to-product formulas to find the exact value of the following expression. cos 120◦ + cos 60◦ = J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Example Use the sum-to-product formulas to find the exact value of the following expression. 120◦ + 60◦ 120◦ − 60◦ cos 120◦ + cos 60◦ = 2 cos cos 2 2 ◦ ◦ = 2 cos (90 ) cos (30 ) = 0 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Application The range of a projectile fired at an angle θ with respect to the horizontal and with an initial velocity of v0 feet per second is r= 1 2 v sin 2θ. 32 0 If an athlete throws a javelin at 75 feet per second, at what angle must the javelin be thrown so that it travels 130 feet? J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Solution 1 (75)2 sin 2θ 32 4160 = 5625 sin 2θ 130 = 0.739556 = sin 2θ 2θ = arcsin(0.739556) = 0.83241 ≈ 47.69◦ θ ≈ 23.85◦ J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas Homework Read Section 5.5. Exercises: 1, 5, 9, 13, . . . , 133, 137 J. Robert Buchanan Multiple-Angle and Product-to-Sum Formulas