Examples of Sum to Product Formulas
Example 1: Evaluate cos 155Β° β cos 25Β°.
Solution:
We know that,
- cos A β cos B = 2 sin[(A+B)/2] sin[(A-B)/2]
cos 155Β° β cos 25Β° = 2 sin [(155Β°+25Β°)/2] sin [(155Β°-25Β°)/2]
β cos 155Β° β cos 25Β° = 2 sin [180Β°/2] sin [130Β°/2]
β cos 155Β° β cos 25Β° = 2 Γ sin 90Β° Γ sin 65Β°
β cos 155Β° β cos 25Β° = 2 Γ 1 Γ 0.9
Thus, cos 155Β° β cos 25Β° = 1.8
Example 2: Solve [sin 4a + sin 2a] /cos a.
Solution:
We know that,
- sin A + sin B = 2 sin[(A+B)/2] cos[(A-B)/2]
sin 4a + sin 2a = 2 sin[(4a+2a)/2] cos[(4a-2a)/2]
β sin 4a + sin 2a = 2 sin[6a/2] cos[2a/2]
β sin 4a + sin 2a = 2 sin 3a cos a
β [sin 4a + sin 2a] /cos a = [2 sin 3a cos a] /cos a
Thus, [sin 4a + sin 2a] /cos a = 2 sin 3a
Example 3: Prove that [sin 10x β sin 4x] / [cos 12x β cos 6x] = cos 7x / sin 9x
Solution:
We know that,
- sin A β sin B = 2 cos[(A+B)/2] sin[(A-B)/2]
- cos A β cos B = 2 sin[(A+B)/2] sin[(A-B)/2]
sin 10x β sin 4x = 2 cos [(10x + 4x)/2] sin [(10x β 4x)/2]
sin 10x β sin 4x = 2 cos [14x/2] sin [6x/2]
β sin 10x β sin 4x = 2 cos 7x sin 3x
β cos 12x β cos 6x = 2 sin [(12x + 6x)/2] sin [(12x β 6x)/2]
β cos 12x β cos 6x = 2 sin [18x/2] sin [6x/2]
β cos 12x β cos 6x = 2 sin 9x sin 3x
β [sin 10x β sin 4x] / [cos 12x β cos 6x] = [2 cos 7x sin 3x] / [2 sin 9x sin 3x]
Thus, [sin 10x β sin 4x] / [cos 12x β cos 6x] = cos 7x / sin 9x
Example 4: Find cos 15x + cos 3x
Solution:
We know that,
- cos A + cos B = 2 cos[(A+B)/2] cos[(A-B)/2]
cos 15x + cos 3x = 2 cos [(15x + 3x)/2] cos [(15x β 3x)/2]
β cos 15x + cos 3x = 2 cos [18x/2] cos [12x/2]
Thus, cos 15x + cos 3x = 2 cos 9x cos 6x
Sum to Product Formulas
The sum to product formulas are trigonometric identities that convert the sum or difference of two trigonometric functions into a product of trigonometric functions. These formulas are particularly useful in simplifying expressions, solving trigonometric equations, and integrating functions.
Sum to Product formulas are important formulas of trigonometry. Four sum-to-product formulas in trigonometry are,
- sin A + sin B = 2 sin [(A+B)/2] Γ cos [(A-B)/2]
- sin A β sin B = 2 cos[(A+B)/2] Γ sin[(A-B)/2]
- cos A + cos B = 2 cos[(A+B)/2] Γ cos[(A-B)/2]
- cos A β cos B = 2 sin[(A+B)/2] Γ sin[(A-B)/2]
In this article, we will learn about Sum to Product Formulas, Proof of Sum to Product Formulas, Application of Sum to Product Formulas in detail.
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