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Multiple Angle & Product-to-Sum Formulas MATH 109 - Precalculus S. Rook Overview • Section 5.5 in the textbook: – Double-angle formulas – Power-reducing formulas – Half-angle formulas – Product-to-sum & sum-to-product formulas 2 Double-Angle Formulas Double-Angle Formulas • Often we are concerned with manipulating an angle of the form ku where k is an integer • We manipulate angles of the form 2u (called doubleangles) so often that we have derived formulas for their usage: – Can be derived from sum and difference formulas – Formulas for angles with multiples other than 2 can also be derived using the sum and difference formulas cos 2u cos 2 u sin 2 u sin 2u 2 sin u cos u 2 tan u tan 2u 1 tan 2 u 2 cos 2 u 1 1 2 sin u 2 4 Double-Angle Formulas (Example) 4 Ex 1: If sin x , x Q IV , find: 5 a) cos 2x b) csc 2x c) tan 2x 5 Double-Angle Formulas (Example) Ex 2: Use the double-angle formulas to find the values of sin 120°, cos 120°, and tan 120° 6 Power-Reducing Formulas Power-Reducing Formulas • Sometimes, we need to manipulate a trigonometric function of the form sinnu, cosnu, tannu • We do this by utilizing the power-reducing formulas – Can be derived from the double-angle formulas • These formulas allow us to write a trigonometric function with power n into cosine functions with a lower power 1 cos 2u 1 cos 2u 1 cos 2u 2 2 2 cos u tan u sin u 2 1 cos 2u 2 8 Power-Reducing Formulas (Example) Ex 3: Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine: cos 4 x 9 Half-Angle Formulas Half-Angle Formulas • Sometimes we are concerned with manipulating an angle of the form 1⁄2u called half-angles: – Formulas can be derived from the power-reducing formulas u u sin and cos – The sign for 2 2 is determined by which quadrant u⁄2 terminates u 1 cos u u 1 cos u cos sin 2 2 2 2 u sin u tan 2 1 cos u 11 Half-Angle Formulas (Example) 5 3 Ex 4: If csc u , u , find: 3 2 a) sin u⁄2 b) cos u⁄2 c) tan u⁄2 12 Product-to-Sum & Sum-toProduct Formulas Product-to-Sum & Sum-to-Product Formulas • The preceding formulas can be used when we have one angle • However, situations arise where we wish to operate on two DIFFERENT angles – e.g. Products such as sin u cos v transform to sums – e.g. Sums such as sin u + cos v transform to products • When considering sines & cosines and two different angles, we have four different situations that can arise 14 Product-to-Sum Formulas • Product-to-Sum Formulas: 1 sin u sin v cosu v cosu v 2 1 cos u cos v cosu v cosu v 2 1 sin u cos v sin u v sin u v 2 1 cos u sin v sin u v sin u v 2 15 Sum-to-Product Formulas • Sum-to-Product Formulas: u v u v sin u sin v 2 sin cos 2 2 u v u v sin u sin v 2 cos sin 2 2 u v u v cos u cos v 2 cos cos 2 2 u v u v cos u cos v 2 sin sin 2 2 16 Product-to-Sum Formulas (Example) Ex 5: Use the product-to-sum formulas to write the product as a sum or difference: 10 cos 75 cos15 17 Sum-to-Product Formulas (Example) Ex 6: Use the sum-to-product formulas to write the sum or difference as a product: sin 5 sin 3 18 Summary • After studying these slides, you should be able to: – Use the double-angle, power-reducing, half-angle, product-to-sum, and sum-to-product formulas to solve problems • Additional Practice – See the list of suggested problems for 5.5 • Next lesson – Solving Trigonometric Equations (Section 5.3) 19