Trigonometric Identities Notes

Reciprocal Identities

  • sin x = 1 / \,

  • csc x = 1 / sin x

  • cos x = 1 / sec x

  • sec x = 1 / cos x

  • tan x = 1 / cot x

  • cot x = 1 / tan x

Notes:

  • These identities express each primary function as the reciprocal of its co-function (csc, sec, cot).

  • They are useful for converting expressions to or from sine, cosine, and tangent forms when solving equations or simplifying expressions.

Pythagorean Identities

  • sin^2 x + cos^2 x = 1

    • can be rearranged as cos^2 x = 1 - sin^2 x and sin^2 x = 1 - cos^2 x

    • equivalent forms: sin^2 x = 1 - cos^2 x, cos^2 x = 1 - sin^2 x

  • sec^2 x - tan^2 x = 1

    • can be rearranged as tan^2 x = sec^2 x - 1 and sec^2 x = tan^2 x + 1

  • csc^2 x - cot^2 x = 1

    • can be rearranged as cot^2 x = csc^2 x - 1 and csc^2 x = cot^2 x + 1

  • Basic ratio forms:

    • tan x = sin x / cos x

    • cot x = cos x / sin x

Significance:

  • These identities link sine and cosine to the reciprocal or reciprocal-trig functions, and they underpin many integrations and simplifications in trigonometry and calculus.

Sum and Difference Identities

  • sin(x + y) = sin x cos y + cos x sin y

  • sin(x - y) = sin x cos y - cos x sin y

  • cos(x + y) = cos x cos y - sin x sin y

  • cos(x - y) = cos x cos y + sin x sin y

Use:

  • Build larger-angle expressions from known angles

  • Prove other trigonometric identities

  • Solve trigonometric equations by expanding or condensing angles

Double-angle Identities

  • sin(2x) = 2 sin x cos x

  • cos(2x) = cos^2 x - sin^2 x

    • can also be written as cos(2x) = 2 cos^2 x - 1

    • or cos(2x) = 1 - 2 sin^2 x

  • tan(2x) = 2 tan x / (1 - tan^2 x)

  • tan(x + y) = (tan x + tan y) / (1 - tan x tan y)

  • tan(x - y) = (tan x - tan y) / (1 + tan x tan y)

Summary:

  • Double-angle identities express trig functions at 2x in terms of x, useful for simplifying integrals and solving equations.

Power-Reducing (Power-Reducing) Identities

  • sin^2 x = (1 - cos(2x)) / 2

  • cos^2 x = (1 + cos(2x)) / 2

  • tan^2 x = (1 - cos(2x)) / (1 + cos(2x))

Notes:

  • These reduce powers of sine and cosine to first powers with a cosine of a double angle, useful in integration and algebraic manipulation.

Half-Angle Identities

  • sin^2(x/2) = (1 - cos x) / 2

  • cos^2(x/2) = (1 + cos x) / 2

  • tan^2(x/2) = (1 - cos x) / (1 + cos x)

Explanation:

  • These identities relate trig functions of half-angles to the full angle, widely used in integration, Fourier analysis, and simplifying expressions.

Product-to-Sum Identities

  • sin x cos y = (1/2) [ sin(x + y) + sin(x - y) ]

  • cos x cos y = (1/2) [ cos(x + y) + cos(x - y) ]

  • sin x sin y = (1/2) [ cos(x - y) - cos(x + y) ]

  • tan x tan y = [ cos(x - y) - cos(x + y) ] / [ cos(x - y) + cos(x + y) ]

Use:

  • Convert products to sums for easier integration or simplification

  • Derive Fourier series, perform trigonometric integrations, or solve equations involving products of trigonometric functions

Even-Odd Identities

  • sin(-x) = - sin x (odd)

  • cos(-x) = cos x (even)

  • tan(-x) = - tan x (odd)

  • cot(-x) = - cot x (odd)

  • sec(-x) = sec x (even)

Notes:

  • Identify symmetry properties of trigonometric functions about the origin and unit circle.

Sum-to-Product Identities

  • sin x + sin y = 2 sin((x + y)/2) cos((x - y)/2)

  • cos x + cos y = 2 cos((x + y)/2) cos((x - y)/2)

  • tan x + tan y does not have a standard simple sum-to-product form; commonly, related expressions are derived from sin, cos combinations.

Use:

  • Transform sums into products to simplify factoring, integrals, or solving equations

Difference-to-Product Identities

  • sin x - sin y = 2 cos((x + y)/2) sin((x - y)/2)

  • cos x - cos y = -2 sin((x + y)/2) sin((x - y)/2)

Notes:

  • Useful for transforming differences of sines/cosines into products, aiding in integration or simplification.

Additional Notes and Conventions

  • csc(-x) = - csc x (cosecant is odd)

  • Domain considerations: denominators cannot be zero where the expressions are defined (e.g., sin x ≠ 0 for cot and csc, cos x ≠ 0 for tan-related forms, etc.).

  • When using these identities, always check for restrictions on x and y for the given problem context.

Connections to foundational principles:

  • These identities arise from definitions of sine and cosine on the unit circle, basic algebraic manipulations, and symmetry properties.

  • They underpin many applications in physics, engineering, signal processing, and computer science, including wave superposition, Fourier analysis, and solving trigonometric equations.

Examples and quick references:

  • Convert tan in terms of sin and cos: \tan x = \frac{\sin x}{\cos x}

  • Convert cot in terms of sin and cos: \cot x = \frac{\cos x}{\sin x}

  • Express Pythagorean identity variants: \sin^2 x + \cos^2 x = 1 \quad\Rightarrow\quad \cos^2 x = 1 - \sin^2 x, \quad \sin^2 x = 1 - \cos^2 x

  • Double-angle example: \sin(2x) = 2 \sin x \cos x, \quad \cos(2x) = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x

  • Product-to-sum example: \sin x \cos y = \tfrac{1}{2} [\sin(x+y) + \sin(x-y)], \quad \cos x \cos y = \tfrac{1}{2} [\cos(x+y) + \cos(x-y)], \quad \sin x \sin y = \tfrac{1}{2} [\cos(x-y) - \cos(x+y)]

Note: Some lines in the provided transcript contained typos. Where standard identities exist, the conventional forms are given here for clarity and exam readiness.