Math Formula for Trigonometry

$Math Formula for Trigonometry - Formula Quest$

Math Formulas for Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It's essential for many fields, including physics, engineering, and astronomy. Below are some fundamental trigonometric formulas and their examples.

1. Basic Trigonometric Ratios

The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). They are defined as follows for a right triangle:

Sine (sin): sin θ = opposite / hypotenuse
Cosine (cos): cos θ = adjacent / hypotenuse
Tangent (tan): tan θ = opposite / adjacent

Example:
For a right triangle with an angle θ = 30°, opposite side = 1, and hypotenuse = 2:

sin 30° = 1 / 2 = 0.5
cos 30° = √3 / 2 ≈ 0.866
tan 30° = 1 / √3 ≈ 0.577

2. Reciprocal Trigonometric Ratios

Cosecant (csc): csc θ = 1 / sin θ
Secant (sec): sec θ = 1 / cos θ
Cotangent (cot): cot θ = 1 / tan θ

Example:
For θ = 30°:

csc 30° = 1 / sin 30° = 2
sec 30° = 1 / cos 30° ≈ 1.155
cot 30° = 1 / tan 30° ≈ 1.732

3. Pythagorean Identities

sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = csc² θ

Example:
For θ = 45°:

sin² 45° + cos² 45° = (√2 / 2)² + (√2 / 2)² = 1/2 + 1/2 = 1

4. Angle Sum and Difference Identities

sin (A + B) = sin A cos B + cos A sin B
sin (A - B) = sin A cos B - cos A sin B
cos (A + B) = cos A cos B - sin A sin B
cos (A - B) = cos A cos B + sin A sin B
tan (A + B) = (tan A + tan B) / (1 - tan A tan B)
tan (A - B) = (tan A - tan B) / (1 + tan A tan B)

Example:
For A = 30° and B = 45°:

sin (30° + 45°) = sin 75° = sin 30° cos 45° + cos 30° sin 45° = 0.5 × √2 / 2 + √3 / 2 × √2 / 2 = √2 / 4 + √6 / 4 = (√2 + √6) / 4

5. Double Angle Identities

sin 2A = 2 sin A cos A
cos 2A = cos² A - sin² A
cos 2A = 2 cos² A - 1
cos 2A = 1 - 2 sin² A
tan 2A = 2 tan A / (1 - tan² A)

Example:
For A = 30°:

sin 2(30°) = sin 60° = 2 sin 30° cos 30° = 2 × 0.5 × √3 / 2 = √3 / 2
cos 2(30°) = cos 60° = 2 cos² 30° - 1 = 2 × (√3 / 2)² - 1 = 2 × 3/4 - 1 = 3/2 - 1 = 0.5

6. Half Angle Identities

sin (A/2) = ±√((1 - cos A) / 2)
cos (A/2) = ±√((1 + cos A) / 2)
tan (A/2) = ±√((1 - cos A) / (1 + cos A))
tan (A/2) = sin A / (1 + cos A)
tan (A/2) = (1 - cos A) / sin A

Example:
For A = 60°:

sin (60° / 2) = sin 30° = √((1 - cos 60°) / 2) = √((1 - 0.5) / 2) = √(0.5 / 2) = √0.25 = 0.5
cos (60° / 2) = cos 30° = √((1 + cos 60°) / 2) = √((1 + 0.5) / 2) = √(1.5 / 2) = √0.75 = √3 / 2

These formulas are the foundation of trigonometry and are used extensively in various applications. Understanding and mastering these will provide a solid base for further studies in mathematics and related fields.