Trigonometric Formulas for Compound Angles and Double Angles
This page delves into more advanced trigonometric formulas, focusing on compound angles (sum and difference of angles) and double angles. These formulas are essential for solving complex trigonometric problems and are widely used in calculus and physics.
The section begins with the formulas for the sum of angles:
- sin(α + β) = sin α cos β + cos α sin β
- cos(α + β) = cos α cos β - sin α sin β
- tan(α + β) = (tan α + tan β) / (1 - tan α tan β)
Following this, the formulas for the difference of angles are presented:
- sin(α - β) = sin α cos β - cos α sin β
- cos(α - β) = cos α cos β + sin α sin β
- tan(α - β) = (tan α - tan β) / (1 + tan α tan β)
Highlight: These compound angle formulas are fundamental in trigonometry and have numerous applications in physics and engineering.
The page then introduces double angle formulas:
- sin(2α) = 2 sin α cos α
- cos(2α) = cos²α - sin²α = 2cos²α - 1 = 1 - 2sin²α
- tan(2α) = 2tan α / (1 - tan²α)
Example: To find sin 60°, you can use the double angle formula with α = 30°: sin 60° = 2 sin 30° cos 30° = 2 * (1/2) * (√3/2) = √3/2
The document also covers half-angle formulas, which are derived from the double angle formulas:
- sin(α/2) = ±√((1 - cos α) / 2)
- cos(α/2) = ±√((1 + cos α) / 2)
- tan(α/2) = ±√((1 - cos α) / (1 + cos α)) = sin α / (1 + cos α) = (1 - cos α) / sin α
These formulas are particularly useful when dealing with angles that are half of common angles, such as 22.5° or 15°.
The page concludes with an introduction to the sine and cosine laws, which are crucial for solving non-right triangles:
Definition: The Law of Sines states that for any triangle ABC, a/sin A = b/sin B = c/sin C, where a, b, and c are the lengths of the sides opposite to angles A, B, and C respectively.
Definition: The Law of Cosines states that for any triangle ABC, a² = b² + c² - 2bc cos A, where a is the side opposite to angle A, and b and c are the other two sides.
These laws are fundamental in trigonometry and have wide-ranging applications in surveying, navigation, and physics.
