Trigonometry formula
Trigonometry formulas are mathematical equations that define the relationships between the sides, angles, and functions of triangles (right-angled and oblique) and periodic phenomena. From high school math exams (SAT, ACT, AP Math) to college-level calculus, engineering, and physics, trigonometry formulas are foundational tools for solving real-world problems—like calculating the height of a skyscraper, predicting wave patterns, or designing GPS systems.
This page is your one-stop reference for all trigonometry formulas—organized by category, explained with examples, and optimized for quick recall. Whether you’re a student cramming for an exam or a professional needing a quick trig formula lookup, we’ve got you covered.
Core Trigonometry Formulas
1. Fundamental Trigonometric Ratios (Right-Angled Triangles)
The “big three” trigonometry formulas (remembered via SOH, CAH, TOA) are the backbone of basic trigonometry:
| Function | Formula (SOH, CAH, TOA) | Reciprocal Formula |
|---|---|---|
| Sine (sin θ) | Opposite / Hypotenuse (SOH) | csc θ = 1 / sin θ |
| Cosine (cos θ) | Adjacent / Hypotenuse (CAH) | sec θ = 1 / cos θ |
| Tangent (tan θ) | Opposite / Adjacent (TOA) | cot θ = 1 / tan θ |
Example:In a right triangle with opposite side = 3, hypotenuse = 5, sin θ = 3/5 = 0.6. Its reciprocal, csc θ = 5/3 ≈ 1.67.
2. Pythagorean Trigonometry Identities
These core trig formulas derive from the Pythagorean theorem and hold true for all valid angles (θ):
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Simplification Example:Solve (1 – cos²θ) × cscθ.Using sin²θ = 1 – cos²θ:= sin²θ × (1/sinθ) = sinθ.
3. Trigonometry Formulas for Complementary & Supplementary Angles
Complementary angles (sum to 90°/π/2) and supplementary angles (sum to 180°/π) follow these key formulas:
Complementary Angles (θ + 90° = 90°)
- sin(90° – θ) = cos θ
- cos(90° – θ) = sin θ
- tan(90° – θ) = cot θ
Supplementary Angles (θ + 180° – θ = 180°)
- sin(180° – θ) = sin θ
- cos(180° – θ) = –cos θ
- tan(180° – θ) = –tan θ
4. Double-Angle & Half-Angle Trigonometry Formulas (H3)
Critical for intermediate/advanced trigonometry:
Double-Angle Formulas
- sin(2θ) = 2 sinθ cosθ
- cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ
- tan(2θ) = (2 tanθ) / (1 – tan²θ)
Half-Angle Formulas
- sin(θ/2) = ±√[(1 – cosθ)/2]
- cos(θ/2) = ±√[(1 + cosθ)/2]
- tan(θ/2) = ±√[(1 – cosθ)/(1 + cosθ)] = (1 – cosθ)/sinθ = sinθ/(1 + cosθ)
5. Sum & Difference Trigonometry Formulas
Use these to calculate trig values for combined angles:
- sin(A ± B) = sinA cosB ± cosA sinB
- cos(A ± B) = cosA cosB ∓ sinA sinB
- tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
6. Law of Sines & Cosines (Oblique Triangles)
For non-right-angled triangles (sides a, b, c opposite angles A, B, C):
Law of Sines
a / sinA = b / sinB = c / sinC
Law of Cosines
- c² = a² + b² – 2ab cosC
- a² = b² + c² – 2bc cosA
- b² = a² + c² – 2ac cosB
Real-World Example:A surveyor needs to find the length of a mountain slope (side c) with a = 500m, b = 700m, and angle C = 60°.Using cosine formula:c² = 500² + 700² – 2(500)(700)cos(60°)c² = 250,000 + 490,000 – 700,000(0.5) = 490,000c = 700m.
Trigonometry Formulas for Graphs
Trigonometric functions are periodic—use these formulas to define their graphs:
| Function | Amplitude | Period | Standard Form |
|---|---|---|---|
| sinθ / cosθ | 1 | 2π | y = A sin(Bθ + C) + D / y = A cos(Bθ + C) + D |
| tanθ | N/A | π | y = A tan(Bθ + C) + D |
Key Variables:
- A: Amplitude (vertical stretch/compression)
- B: Period = 2π/|B| (for sin/cos) or π/|B| (for tan)
- C: Phase Shift = –C/B (horizontal shift)
- D: Vertical Shift (up/down)
Example:Graph y = 2 sin(3θ – π/2) + 1
- Amplitude = 2
- Period = 2π/3
- Phase Shift = π/6 (right)
- Vertical Shift = 1 (up)
Inverse Trigonometry Formulas
Inverse trig functions (arcsin, arccos, arctan) reverse the output of standard trig functions:
- sin(arcsin x) = x (for –1 ≤ x ≤ 1)
- cos(arccos x) = x (for –1 ≤ x ≤ 1)
- tan(arctan x) = x (for all real x)
Additional Inverse Formulas:
- arcsin x + arccos x = π/2
- arctan x + arccot x = π/2
Trigonometry Formula Cheat Sheet
Save time with our printable cheat sheet:
Practice Problems Using Trigonometry Formulas
- Trigonometry Charts(33+)
- Quadrants and Angles(18+)
- Convert between Degrees and Radians(18+)
- Degrees, Minutes and Seconds(15+)
- Reference and Coterminal Angles(18+)
- Trigonometric Ratios(78+)
- Trigonometric Identities(105+)
- Unit Circle(21+)
- Trigonometric Ratios of Allied Angles(12+)
- Evaluating Trigonometric Expressions(18+)
- Evaluating Trigonometric Functions(5+)
- Inverse Trigonometric Functions(21+)
- Law of Sines(21+)
- Law of Cosines(18+)
- Solving Triangles(12+)
- Principal Solutions of Trigonometric Equations(15+)
- General Solutions of Trigonometric Equations(15+)
FAQs About Trigonometry Formulas
Q1: What is the most important trigonometry formula?
A1: The Pythagorean identity (sin²θ + cos²θ = 1) is the most versatile—it’s used to simplify almost all other trig formulas and solve equations.
Q2: How do I memorize trigonometry formulas fast?
A2: Group formulas by category (e.g., ratios, identities, angles), use mnemonics (SOHCAHTOA), and practice applying them to problems daily. Our cheat sheet also helps with quick recall.
Q3: Are trigonometry formulas the same for radians and degrees?
A3: The formulas themselves are identical—only the angle units change. For example, sin(90°) = sin(π/2) = 1.
Q4: Where are trigonometry formulas used in real life?
A4: Engineers use them for structural design, pilots for navigation, astronomers for calculating celestial distances, and game developers for 3D graphics.
Conclusion (H2)
Trigonometry formulas are not just abstract equations—they’re practical tools that power countless industries and academic pursuits. This guide organizes every essential trigonometry formula in one easy-to-access place, with examples and practice to help you apply them confidently.
Whether you’re studying for an exam, working on a project, or just need a quick reference, our trigonometry formula guide is your go-to resource. Download our cheat sheet, practice the problems, and master trigonometry once and for all!