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All Trigonometry Formulas PDF

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All Trigonometry Formulas PDF

Trigonometry is an essential branch of mathematics that focuses on the relationships between the angles and sides of triangles. For students, teachers, and professionals alike, having all trigonometry formulas in one place is extremely helpful. Whether preparing for an exam or solving complex problems, these formulas make the process more efficient. While some people may prefer downloadable resources, this topic provides a complete and accessible list of all trigonometry formulas without needing a PDF. It is written in a way that is easy to understand and designed to help anyone grasp key trigonometric concepts.

Basic Trigonometric Ratios

Primary Trigonometric Functions

The three main trigonometric functions relate the angles of a right triangle to the ratios of its sides:

  • Sine (sin)= Opposite / Hypotenuse
  • Cosine (cos)= Adjacent / Hypotenuse
  • Tangent (tan)= Opposite / Adjacent

Reciprocal Trigonometric Functions

These are the reciprocals of the primary functions:

  • Cosecant (csc)= 1 / sin = Hypotenuse / Opposite
  • Secant (sec)= 1 / cos = Hypotenuse / Adjacent
  • Cotangent (cot)= 1 / tan = Adjacent / Opposite

Pythagorean Identities

These identities are derived from the Pythagorean Theorem and are foundational in trigonometry:

  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = csc²θ

Trigonometric Ratios of Standard Angles

Values for Common Angles (Degrees)

It is important to memorize the trigonometric values for key angles:

  • sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1
  • cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0
  • tan 0° = 0, tan 30° = √3/3, tan 45° = 1, tan 60° = √3, tan 90° = undefined

Trigonometric Formulas for Sum and Difference

Angle Sum and Difference Identities

These formulas help find the trigonometric function of the sum or difference of two angles:

  • sin(A ± B) = sin A cos B ± cos 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)

Double Angle and Triple Angle Formulas

Double Angle Formulas

  • sin(2A) = 2 sin A cos A
  • cos(2A) = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A
  • tan(2A) = 2 tan A / (1 – tan²A)

Triple Angle Formulas

  • sin(3A) = 3 sin A – 4 sin³A
  • cos(3A) = 4 cos³A – 3 cos A
  • tan(3A) = (3 tan A – tan³A) / (1 – 3 tan²A)

Product to Sum and Sum to Product Formulas

Product to Sum

  • sin A sin B = ½ [cos(A – B) – cos(A + B)]
  • cos A cos B = ½ [cos(A – B) + cos(A + B)]
  • sin A cos B = ½ [sin(A + B) + sin(A – B)]

Sum to Product

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

Inverse Trigonometric Formulas

Basic Inverse Identities

These formulas involve the inverse functions of trigonometric ratios:

  • sin⁻¹(sin x) = x, for x in [–Ï€/2, π/2]
  • cos⁻¹(cos x) = x, for x in [0, π]
  • tan⁻¹(tan x) = x, for x in (–Ï€/2, π/2)

Other Useful Inverse Formulas

  • sin⁻¹x + cos⁻¹x = π/2
  • tan⁻¹x + cot⁻¹x = π/2
  • sec⁻¹x = cos⁻¹(1/x), for |x| ≥ 1
  • csc⁻¹x = sin⁻¹(1/x), for |x| ≥ 1

Trigonometric Equations and Solutions

General Solutions for Common Functions

These are the general solutions for equations involving sine, cosine, and tangent:

  • sin θ = sin α ⇒ θ = nπ + (–1)ⁿα
  • cos θ = cos α ⇒ θ = 2nπ ± α
  • tan θ = tan α ⇒ θ = nπ + α

Trigonometry in Coordinate Geometry

Functions in Terms of Coordinates

In a coordinate system, trigonometric ratios can also be expressed using coordinates (x, y) and the distance r from the origin:

  • sin θ = y / r
  • cos θ = x / r
  • tan θ = y / x
  • r = √(x² + y²)

Trigonometry in Different Quadrants

The sign of trigonometric functions changes depending on the quadrant of the angle:

  • First quadrant: All functions are positive
  • Second quadrant: sin and csc are positive
  • Third quadrant: tan and cot are positive
  • Fourth quadrant: cos and sec are positive

Law of Sines and Law of Cosines

Law of Sines

Used for solving any triangle (not just right-angled):

  • (sin A)/a = (sin B)/b = (sin C)/c

Law of Cosines

Helpful when we know two sides and an included angle or all three sides:

  • c² = a² + b² – 2ab cos C
  • a² = b² + c² – 2bc cos A
  • b² = a² + c² – 2ac cos B

Applications of Trigonometry

Real-World Uses

Trigonometry formulas are used in many real-life fields, including:

  • Engineering and Architecture
  • Astronomy and Navigation
  • Physics and Wave Theory
  • Surveying and Mapping

Understanding these formulas allows students and professionals to solve practical problems involving heights, distances, angles, and more.

This collection of all trigonometry formulas provides a useful reference for learners at any level. Rather than relying on a PDF, students can use this guide to understand and apply trigonometric principles in real time. From identities and ratios to angle transformations and coordinate geometry, mastering these formulas helps build a strong mathematical foundation. Regular practice and application of these trigonometric identities and rules will improve problem-solving skills and boost confidence in mathematics.