Additional Identities Trigonometry MATH 103 S. Rook

Overview Section 5.5 in the textbook: – Identities and formulas involving inverse functions – Product to sum & sum to product formulas 2

Identities and Formulas Involving Inverse Trigonometric Functions

We have discussed how to solve problems involving the inverse trigonometric functions: – e.g. cos(arcsin ½) – Draw and label a right triangle to solve Possible for inverse trigonometric functions to appear as arguments in the sum & difference formulas, double-angle formulas, or half-angle formulas – e.g. tan(arccos -½ – arcsin ½) Let A = arccos -½ and B = arcsin ½ Becomes tan(A – B) → difference formula for the tangent 4

Identities and Formulas Involving Inverse Trigonometric Functions (Example) Ex 1: Evaluate without a calculator: a) b) 5

Product to Sum & Sum to Product 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 A cos B transform to sums – e.g. Sums such as sin A + cos B transform to products When considering sines & cosines and two different angles, we have four different situations that can arise 7

Product-to-Sum Formulas Product-to-Sum Formulas: 8

Sum-to-Product Formulas Sum-to-Product Formulas: 9

Product to Sum Formulas (Example) Ex 2: Express as a sum or difference: 10

Sum to Product Formulas (Example) Ex 3: Express as a product: 11

Summary After studying these slides, you should be able to: – Apply the different types of formulas learned in Chapter 5 to the inverse trigonometric functions – Apply the product to sum formulas – Apply the sum to product formulas Additional Practice – See the list of suggested problems for 5.5 Next lesson – Solving Trigonometric Equations (Section 6.1) 12