Double-Angle and Half-Angle Formulas The slides for this text are organized into chapters. This lecture covers Chapter 1. Chapter 1: Introduction to Database Systems Chapter 2: The Entity-Relationship Model Chapter 3: The Relational Model Chapter 4 (Part A): Relational Algebra Chapter 4 (Part B): Relational Calculus Chapter 5: SQL: Queries, Programming, Triggers Chapter 6: Query-by-Example (QBE) Chapter 7: Storing Data: Disks and Files Chapter 8: File Organizations and Indexing Chapter 9: Tree-Structured Indexing Chapter 10: Hash-Based Indexing Chapter 11: External Sorting Chapter 12 (Part A): Evaluation of Relational Operators Chapter 12 (Part B): Evaluation of Relational Operators: Other Techniques Chapter 13: Introduction to Query Optimization Chapter 14: A Typical Relational Optimizer Chapter 15: Schema Refinement and Normal Forms Chapter 16 (Part A): Physical Database Design Chapter 16 (Part B): Database Tuning Chapter 17: Security Chapter 18: Transaction Management Overview Chapter 19: Concurrency Control Chapter 20: Crash Recovery Chapter 21: Parallel and Distributed Databases Chapter 22: Internet Databases Chapter 23: Decision Support Chapter 24: Data Mining Chapter 25: Object-Database Systems Chapter 26: Spatial Data Management Chapter 27: Deductive Databases Chapter 28: Additional Topics Dr .Hayk Melikyan Departmen of Mathematics and CS melikyan@nccu.edu

Double-Angle Identities

Three Forms of the Double-Angle Formula for cos2

Power-Reducing Formulas

Example Write an equivalent expression for sin4x that does not contain powers of trigonometric functions greater than 1. Solution

Half-Angle Identities

Text Example Find the exact value of cos 112.5°. Solution Because 112.5° = 225°/2, we use the half-angle formula for cos /2 with  = 225°. What sign should we use when we apply the formula? Because 112.5° lies in quadrant II, where only the sine and cosecant are positive, cos 112.5° < 0. Thus, we use the - sign in the half-angle formula.

Half-Angle Formulas for:

Verifying a Trigonometric Identity Verify the identity:

The Half-Angle Formulas for Tangent

Verify the following identity: Example Verify the following identity: Solution

Product-to-Sum and Sum-to-Product Formulas Product-to-Sum Formulas

Example Express the following product as a sum or difference: Solution

Text Example Express each of the following products as a sum or difference. a. sin 8x sin 3x b. sin 4x cos x Solution The product-to-sum formula that we are using is shown in each of the voice balloons. a. sin 8x sin 3x = 1/2[cos (8x - 3x) - cos(8x + 3x)] = 1/2(cos 5x - cos 11x) sin  sin  = 1/2 [cos( - ) - cos( + )] sin  cos  = 1/2[sin( + ) + sin( - )] b. sin 4x cos x = 1/2[sin (4x + x) + sin(4x - x)] = 1/2(sin 5x + sin 3x)

Evaluating the Product of a Trigonometric Expression Determine the exact value of the expression

Sum-to-Product Formulas

Express the difference as a product: Example Express the difference as a product: Solution

Express the sum as a product: Example Express the sum as a product: Solution

Verify the following identity: Example Verify the following identity: Solution

Example

Example

Equations Involving a Single Trigonometric Function To solve an equation containing a single trigonometric function: • Isolate the function on one side of the equation. sinx = a (-1 ≤ a ≤ 1 ) cosx = a (-1 ≤ a ≤ 1 ) tan x = a ( for any real a ) • Solve for the variable.

Trigonometric Equations y y = cos x 1 y = 0.5 x –4  –2  2  4  –1 cos x = 0.5 has infinitely many solutions for –  < x <  y y = cos x 1 0.5 x 2  cos x = 0.5 has two solutions for 0 < x < 2 –1

Text Example Solve the equation: 3 sin x - 2 = 5 sin x - 1. Solution The equation contains a single trigonometric function, sin x. Step 1 Isolate the function on one side of the equation. We can solve for sin x by collecting all terms with sin x on the left side, and all the constant terms on the right side. 3 sin x - 2 = 5 sin x - 1 This is the given equation. 3 sin x - 5 sin x - 2 = 5 sin x - 5 sin x – 1 Subtract 5 sin x from both sides. -2 sin x - 2 = -1 Simplify. -2 sin x = 1 Add 2 to both sides. sin x = -1/2 Divide both sides by -2 and solve for sin x.

Text Example Solve the equation: 2 cos2 x + cos x - 1 = 0, 0 £ x < 2p. Solution The given equation is in quadratic form 2t2 + t - 1 = 0 with t = cos x. Let us attempt to solve the equation using factoring. 2 cos2 x + cos x - 1 = 0 This is the given equation. (2 cos x - 1)(cos x + 1) = 0 Factor. Notice that 2t2 + t – 1 factors as (t – 1)(2t + 1). 2 cos x - 1= 0 or cos x + 1 = 0 Set each factor equal to 0. 2 cos x = 1 cos x = -1 Solve for cos x. cos x = 1/2 x = p x = 2ppp x = p The solutions in the interval [0, 2p) are p/3, p, and 5p/3.

Solve the following equation: Example Solve the following equation: Solution:

Solve the equation on the interval [0,2) Example Solve the equation on the interval [0,2) Solution:

Solve the equation on the interval [0,2) Example Solve the equation on the interval [0,2) Solution:

Solve the equation on the interval [0,2) Example Solve the equation on the interval [0,2) Solution: