1. 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

2. Double-Angle Identities

3. Three Forms of the Double-Angle Formula for cos2

4. Power-Reducing Formulas

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

6. Half-Angle Identities

7. 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.

8. Half-Angle Formulas for:

9. Verifying a Trigonometric Identity
Verify the identity:

10. The Half-Angle Formulas for Tangent

11. Verify the following identity:
Example
Verify the following identity:
Solution

12. Product-to-Sum and Sum-to-Product Formulas
Product-to-Sum Formulas

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

14. 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)

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

16. Sum-to-Product Formulas

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

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

19. Verify the following identity:
Example
Verify the following identity:
Solution

20. Example

21. Example

22. 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.

23. Trigonometric Equationsy 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

24. 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.

25. 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= 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.

26. Solve the following equation:
Example
Solve the following equation:
Solution:

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

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

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