1.2 Pythagorean Identities
| Type |
Formula |
| Identity 1 |
`\sin^2(\theta) + \cos^2(\theta) = 1` |
| Identity 2 |
`1 + \tan^2(\theta) = \sec^2(\theta)` |
| Identity 3 |
`1 + \cot^2(\theta) = \csc^2(\theta)` |
1.3 Basic Angle Identities
| Identities |
Formula |
| Sine Angle Sum |
`\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)` |
| Sine Angle Difference |
`\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)` |
| Cosine Angle Sum |
`\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)` |
| Cosine Angle Difference |
`\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)` |
| Tangent Angle Sum |
`\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}` |
| Tangent Angle Difference |
`\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}` |
1.4 Complementary Angle Identities
| Complement of: |
Formula |
| Sine |
`\sin(90^\circ - \theta) = \cos(\theta)` |
| Cosine |
`\cos(90^\circ - \theta) = \sin(\theta)` |
| Tangent |
`\tan(90^\circ - \theta) = \cot(\theta)` |
| Cotangent |
`\cot(90^\circ - \theta) = \tan(\theta)` |
| Secant |
`\sec(90^\circ - \theta) = \csc(\theta)` |
| Cosecant |
`\csc(90^\circ - \theta) = \sec(\theta)` |
2. Medium Level (Intermediate Trigonometry Formulas)
2.1 Double Angle Formulas
| Angle |
Formula |
| `\sin(2\theta) =` |
` 2\sin(\theta)\cos(\theta) ` |
| `\cos(2\theta) = ` |
`\cos^2(\theta) - \sin^2(\theta)`,
`2\cos^2(\theta) - 1`,
`1 - 2\sin^2(\theta) ` |
| `\tan(2\theta) = ` |
`\frac{2\tan(\theta)}{1 - \tan^2(\theta)} ` |
2.2 Half Angle Formulas
| Type |
Formula |
| Sine |
`\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}} ` |
| Cosine |
`\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} ` |
| Tangent |
`\tan\left(\frac{\theta}{2}\right) = +-sqrt((1-cos \theta)/(1+cos \theta)) = \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{1 - \cos(\theta)}{\sin(\theta)} ` |
2.3 Product-to-Sum Identities
| Type |
Formula |
| Product-to-Sum Identity 1 |
`\(\sin(a) \sin(b) = \frac{1}{2} [\cos(a - b) - \cos(a + b)]\)` |
| Product-to-Sum Identity 2 |
`\(\cos(a) \cos(b) = \frac{1}{2} [\cos(a - b) + \cos(a + b)]\)` |
| Product-to-Sum Identity 3 |
`\(\sin(a) \cos(b) = \frac{1}{2} [\sin(a + b) + \sin(a - b)]\)` |
2.4 Sum-to-Product Identities
| Type |
Formula |
| Sum-to-Product Identity 1 |
`\(\sin(a) + \sin(b) = 2 \sin\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right)\)` |
| Sum-to-Product Identity 2 |
`\(\sin(a) - \sin(b) = 2 \cos\left(\frac{a + b}{2}\right) \sin\left(\frac{a - b}{2}\right)\)` |
| Sum-to-Product Identity 3 |
`\(\cos(a) + \cos(b) = 2 \cos\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right)\)` |
| Sum-to-Product Identity 4 |
`\(\cos(a) - \cos(b) = -2 \sin\left(\frac{a + b}{2}\right) \sin\left(\frac{a - b}{2}\right)\)` |
2.5 Inverse Trigonometric Relationships
| Type |
Formula |
| Inverse Sine Relationship |
`\(\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}\)` |
| Inverse Tangent Relationship |
`\(\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}\)` |
| Inverse Secant Relationship |
`\(\sec^{-1}(x) + \csc^{-1}(x) = \frac{\pi}{2}\)` |
| Function |
Relationship |
Domain |
| Inverse Sine |
`\(\sin^{-1}(\sin x) = x\)` |
`\(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\)` |
| Inverse Cosine |
`\(\cos^{-1}(\cos x) = x\)` |
`\(0 \leq x \leq \pi\)` |
| Inverse Tangent |
`\(\tan^{-1}(\tan x) = x\)` |
`\(-\frac{\pi}{2} lt x lt \frac{\pi}{2}\)` |
3. Complex and High-Level Trigonometry
3.1 Laws of Trigonometry (Triangles)
| Law |
Formula |
Description |
| Law of Sines |
`\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)` |
Relates the sides of a triangle to the sines of its angles. |
| Law of Cosines |
`\(c^2 = a^2 + b^2 - 2ab \cos C\)` |
Relates the lengths of the sides of a triangle to the cosine of one of its angles. |
| Law of Tangents |
`\(\frac{a - b}{a + b} = \tan\left(\frac{A - B}{2}\right) \tan\left(\frac{A + B}{2}\right)\)` |
Relates the sides of a triangle to the tangents of half of its angles. |
3.2 Advanced Identities
| Identity |
Formula |
| Sine of the Sum of Three Angles |
`\(\sin(A + B + C) = \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C\)` |
| Cosine of the Sum of Three Angles |
`\(\cos(A + B + C) = \cos A \cos B \cos C - \sin A \sin B \cos C - \sin A \cos B \sin C - \cos A \sin B \sin C\)` |
3.3 Euler’s Formula and Complex Trigonometry
| Concept |
Formula | `
| Euler's Formula |
`\(e^{i\theta} = \cos \theta + i \sin \theta\)` |
| Cosine in Exponential Form |
`\(\cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}\)` |
| Sine in Exponential Form |
`\(\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}\)` |
3.4 Hyperbolic Trigonometric Functions
| Concept |
Formula | `
| Hyperbolic Sine |
`\(\sinh x = \frac{e^x - e^{-x}}{2}\)` |
`
| Hyperbolic Cosine |
`\(\cosh x = \frac{e^x + e^{-x}}{2}\)` |
| Hyperbolic Tangent |
`\(\tanh x = \frac{\sinh x}{\cosh x}\)` |
| Pythagorean Identity (Hyperbolic) |
`\(\cosh^2 x - \sinh^2 x = 1\)` |
| Identity |
Formula |
Description |
| Hyperbolic Identity |
` \(1 - \tanh^2 x = \text{sech}^2 x\)` |
Relates hyperbolic tangent and hyperbolic secant functions. |
| Definition of Hyperbolic Tangent |
`\(\tanh x = \frac{\sinh x}{\cosh x}\)` |
Ratio of hyperbolic sine to hyperbolic cosine. |
| Definition of Hyperbolic Secant |
`\(\text{sech} x = \frac{1}{\cosh x}\)` |
Reciprocal of hyperbolic cosine. |
3.5 Integration and Differentiation
| Concept |
Formula | `
| Derivative of Sine |
`\(\frac{d}{dx}[\sin x] = \cos x\)` |
| Derivative of Cosine |
`\(\frac{d}{dx}[\cos x] = -\sin x\)` |
| Integral of Sine |
`\(\int \sin x \, dx = -\cos x + C\)` |
| Integral of Cosine |
`\(\int \cos x \, dx = \sin x + C\)` |
| Integral of Secant Squared |
`\(\int \sec^2 x \, dx = \tan x + C\)` |
3.6 Trigonometric Expansions
These formulas involve expanding products and powers of trigonometric functions.
| Type |
Formula |
| Power Reduction Identity (Sine) |
`\(\sin^2 x = \frac{1 - \cos(2x)}{2}\)` |
| Power Reduction Identity (Cosine) |
`\(\cos^2 x = \frac{1 + \cos(2x)}{2}\)` |
| Power Reduction Identity (Tangent) |
`\(\tan^2 x = \frac{1 - \cos(2x)}{1 + \cos(2x)}\)` |
| Triple Angle Formula (Sine) |
`\(\sin(3x) = 3\sin x - 4\sin^3 x\)` |
| Triple Angle Formula (Cosine) |
`\(\cos(3x) = 4\cos^3 x - 3\cos x\)` |
| Triple Angle Formula (Tangent) |
`\(\tan(3x) = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}\)` |
`
3.7 Cyclic Identities Summary Table
| Type |
Identity |
| Sine Identity |
`\(\sin(A + B) + \sin(A - B) = 2 \sin A \cos B\)` |
| Cosine Identity |
`\(\cos(A + B) + \cos(A - B) = 2 \cos A \cos B\)` |
3.8 Advanced Integration and Differentiation
| Concept |
Formula |
| Reduction Formula (Sine) |
`\(\int \sin^n x \ dx = -\frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x \, dx\)` |
| Reduction Formula (Cosine) |
`\(\int \cos^n x \ dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2} x \, dx\)` |
| Derivative of Inverse Sine |
`\(\frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1\)` |
| Derivative of Inverse Cosine |
`\(\frac{d}{dx}[\cos^{-1} x] = -\frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1\)` |
| Derivative of Inverse Tangent |
`\(\frac{d}{dx}[\tan^{-1} x] = \frac{1}{1 + x^2}\)` |
3.9 Spherical Trigonometry Formulas
| Law |
Formula |
| Spherical Law of Cosines |
`\(\cos c = \cos a \cos b + \sin a \sin b \cos C\)` |
| Spherical Law of Sines |
`\(\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}\)` |
Spherical Trigonometry Formulas
( A, B, C ) are the angles of the spherical triangle.
( a, b, c ) are the lengths of the sides opposite to angles ( A, B, C ), respectively.
| Angles |
Lengths |
| `sin"" 1/2 A = [(sin(s-b)sin(s-c))/ (sin b sin c)]^(1/2)` |
`sin"" 1/2 a = [(-cosS cos(S-A))/ (sin B sin C)]^(1/2)` |
| `cos"" 1/2 A = [(sin s sin(s-a))/ (sin b sin c)]^(1/2)` |
`cos"" 1/2 a = [(cos(S-B) cos(S-C))/ (sin B sin C)]^(1/2)` |
| `tan"" 1/2 A = [(sin(s-b)sin(s-c))/ (sin s sin(s-a))]^(1/2)` |
`tan"" 1/2 a = [(-cosS cos(S-A))/ (sin(S-B) cos(S-C))]^(1/2)` |
3.10 Transformations in Trigonometry Formulas
| Transformation Type |
Formula | `
| Fourier Transform |
`\(f(t) = \int_{-\infty}^{\infty} F(\omega) e^{i \omega t} \, d\omega\)` |
| Inverse Fourier Transform |
`\(F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} \, dt\)` |
| Wave Equation |
`\(y = A \sin(kx - \omega t + \phi)\)` |
3.11 Higher Angle Identities Summary Table
| Identity Type |
Formula |
| Quadruple Angle Formula (Sine) |
`\(\sin(4x) = 8 \sin x \cos^3 x - 4 \sin^3 x \cos x\)` |
| Quadruple Angle Formula (Cosine) |
`\(\cos(4x) = 8 \cos^4 x - 8 \cos^2 x + 1\)` |
| Quadruple Angle Formula (Tangent) |
`\(\tan(4x) = \frac{4 \tan x - 4 \tan^3 x}{1 - 6 \tan^2 x + \tan^4 x}\)` |
3.12 Trigonometric Identities for Negative Angles Summary Table
| Function |
Identity |
| Sine |
`\(\sin(-x) = -\sin x\)` |
| Cosine |
`\(\cos(-x) = \cos x\)` |
| Tangent |
`\(\tan(-x) = -\tan x\)` |
| Cosecant |
`\(\csc(-x) = -\csc x\)` |
| Secant |
`\(\sec(-x) = \sec x\)` |
| Cotangent |
`\(\cot(-x) = -\cot x\)` |
3.13 Complex Argument Identities Summary Table
These are useful for functions of complex numbers or when analyzing periodic functions.
| Function |
Identity |
| Sine |
`\(\sin(a + ib) = \sin a \cosh b + i \cos a \sinh b\)` |
| Cosine |
`\(\cos(a + ib) = \cos a \cosh b - i \sin a \sinh b\)` |
| Tangent |
`\(\tan(a + ib) = \frac{\sin(2a) + i \sinh(2b)}{\cos(2a) + \cosh(2b)}\)` |
3.14 Trigonometric Forms in Geometry
| Concept |
Formula |
| Area of Triangle |
`\(\text{Area} = \frac{1}{2}ab \sin C\)` |
| Circumradius |
`\(R = \frac{abc}{4\Delta}\)` |
| Inradius |
`\(r = \frac{\Delta}{s}\)` |
3.15 Transformations and Scaling Summary Table
| Concept |
Formula |
| Scaling of Sine and Cosine |
`\(A \sin x + B \cos x = R \sin(x + \phi)\)` |
| Resultant Amplitude |
`\(R = \sqrt{A^2 + B^2}\)` |
| Phase Shift |
`\(\tan \phi = \frac{B}{A}\)` |
| Damped Oscillation |
`\(y(t) = A e^{-\alpha t} \sin(\omega t + \phi)\)` |
3.16 Specialized Trigonometric Relations Formula
| Concept |
Formula |
| Barycentric Trigonometry (Sine) |
`\(\sin A = \frac{a}{2R}\)` |
| Barycentric Trigonometry (Cosine) |
`\(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)` |
| Napier's Analogies |
`\(\tan\left(\frac{A - B}{2}\right) = \frac{\sin(A - B)}{\cos(A + B) + \cos(A - B)}\)` |
3.17 Infinite Product and Weierstrass Substitution Formulas
| Concept |
Formula |
| Infinite Product for Sine |
`\(\sin x = x \prod_{n=1}^{\infty} \left(1 - \frac{x^2}{n^2 \pi^2}\right)\)` |
| Weierstrass Substitution |
`\(t = \tan\left(\frac{x}{2}\right)\)` |
| Sine w.r.t t |
`\(\sin x = \frac{2t}{1 + t^2}\)` |
| Cosine w.r.t t |
`\(\cos x = \frac{1 - t^2}{1 + t^2}\)` |
| Differential Transformation |
`\(dx = \frac{2 \, dt}{1 + t^2}\)` |
