45 Essential Trigonometry Formulas

Trigonometry is a branch of mathematics that deals with relationships between the sides and angles of triangles. Mastering trigonometric formulas is crucial for solving problems in various fields, including physics, engineering, and computer science. Here's a comprehensive list of 45 essential trigonometry formulas.

1. Pythagorean Identities (3)

• sin²(θ) + cos²(θ) = 1
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = csc²(θ)

2. Quotient Identities (2)

• tan(θ) = sin(θ) / cos(θ)
• cot(θ) = cos(θ) / sin(θ)

3. Reciprocal Identities (6)

• csc(θ) = 1 / sin(θ)
• sec(θ) = 1 / cos(θ)
• cot(θ) = 1 / tan(θ)
• sin(θ) = 1 / csc(θ)
• cos(θ) = 1 / sec(θ)
• tan(θ) = 1 / cot(θ)

4. Co-function Identities (6)

• sin(90° - θ) = cos(θ)
• cos(90° - θ) = sin(θ)
• tan(90° - θ) = cot(θ)
• cot(90° - θ) = tan(θ)
• sec(90° - θ) = csc(θ)
• csc(90° - θ) = sec(θ)

5. Sum and Difference Identities (6)

• 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))

6. Double-Angle Identities (6)

• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ)
• cos(2θ) = 2cos²(θ) - 1
• cos(2θ) = 1 - 2sin²(θ)
• tan(2θ) = 2tan(θ) / (1 - tan²(θ))
• cot(2θ) = (cot²(θ) - 1) / (2cot(θ))

7. Half-Angle Identities (6)

• sin(θ/2) = ±√((1 - cos(θ)) / 2)
• cos(θ/2) = ±√((1 + cos(θ)) / 2)
• tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ)))
• tan(θ/2) = sin(θ) / (1 + cos(θ))
• tan(θ/2) = (1 - cos(θ)) / sin(θ)
• cot(θ/2) = ±√((1 + cos(θ)) / (1 - cos(θ)))

8. Product-to-Sum Identities (4)

• sin(A)cos(B) = 1/2[sin(A + B) + sin(A - B)]
• cos(A)sin(B) = 1/2[sin(A + B) - sin(A - B)]
• cos(A)cos(B) = 1/2[cos(A + B) + cos(A - B)]
• sin(A)sin(B) = 1/2[cos(A - B) - cos(A + B)]

9. Sum-to-Product Identities (4)

• sin(A) + sin(B) = 2sin((A + B)/2)cos((A - B)/2)
• sin(A) - sin(B) = 2cos((A + B)/2)sin((A - B)/2)
• cos(A) + cos(B) = 2cos((A + B)/2)cos((A - B)/2)
• cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2)

Further Resources

For more detailed explanations and examples, explore these resources on trigonometric identities from pakmath.com:

• Trigonometric Identities
• Pythagorean Identities
• Sum and Difference Identities
• Double and Half Angle Identities

These formulas are fundamental to trigonometry and are used extensively in various mathematical and scientific applications. Mastering them will significantly improve your problem-solving skills.