Trig Identities

The Law of Sines (sinA/a = sinB/b = sinC/c) is explained for solving triangles when a side and its opposite angle are known. The Law of Cosines (c² = a² + b² - 2ab cosC) is presented for finding a missing side or angle when all three sides or two sides and the included angle are known.

The video starts by defining a right triangle and its sides: opposite, adjacent, and hypotenuse, in relation to an angle Theta. It then introduces the fundamental trigonometric ratios using the mnemonic SOH CAH TOA: Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), and Tangent (Opposite/Adjacent).

This section explains reciprocal identities: Cosecant (1/Sine), Secant (1/Cosine), and Cotangent (1/Tangent). It clarifies that these relationships are reversible, meaning Sine is 1/Cosecant, etc.

The video demonstrates how to calculate all six trigonometric ratios for a 3-4-5 right triangle using the SOH CAH TOA and reciprocal identities.

Quotient identities are introduced: Tangent (Sine/Cosine) and Cotangent (Cosine/Sine), highlighting that Cotangent is the reciprocal of Tangent.

The three main Pythagorean identities are detailed: sin²θ + cos²θ = 1. The other two identities (1 + cot²θ = csc²θ and 1 + tan²θ = sec²θ) are derived by dividing the first identity by sin²θ and cos²θ, respectively.

This part explains even and odd trigonometric functions. Sine, Tangent, Cosecant, and Cotangent are identified as odd functions (e.g., sin(-θ) = -sinθ), while Cosine and Secant are even functions (e.g., cos(-θ) = cosθ).

Cofunction identities are presented, showing relationships like cosθ = sin(90° - θ) or cosθ = sin(π/2 - θ). The video illustrates how these identities work by demonstrating that if two angles add up to 90 degrees, their cofunctions are equal.

The double angle identities for Sine (sin2θ = 2sinθcosθ), Cosine (cos2θ has three forms: cos²θ - sin²θ, 2cos²θ - 1, and 1 - 2sin²θ), and Tangent (tan2θ = 2tanθ / (1 - tan²θ)) are provided. It explains how to derive the different forms of cos2θ from the fundamental Pythagorean identity.

Half angle identities for Sine, Cosine, and Tangent are covered. sin(θ/2) = ±√((1 - cosθ)/2), cos(θ/2) = ±√((1 + cosθ)/2). The video also derives alternative forms for tan(θ/2) from its square root form by multiplying by conjugates.

The sum and difference identities for Sine, Cosine, and Tangent are presented, highlighting the sign conventions for each. For sine(α ± β), the signs match; for cosine(α ± β), the signs are opposite; for tangent(α ± β), the numerator signs match, and the denominator signs are opposite.

Power reducing formulas are introduced to convert squared trigonometric functions into first-power functions with doubled angles: sin²θ = (1 - cos2θ)/2, cos²θ = (1 + cos2θ)/2, and tan²θ = (1 - cos2θ)/(1 + cos2θ). The tangent formula is shown to be derivable from the sine and cosine power reducing formulas.

Four product-to-sum formulas are listed, transforming products of trigonometric functions into sums or differences. For example, sinαsinβ = ½[cos(α - β) - cos(α + β)].

Four sum-to-product formulas are provided, converting sums or differences of trigonometric functions into products. For instance, sinα + sinβ = 2sin((α + β)/2)cos((α - β)/2).

The video briefly touches upon calculating the area of a triangle using trigonometry (Area = ½ab sinC) and Heron's formula. It also mentions the less commonly used Law of Tangents, noting its complexity compared to the Laws of Sines and Cosines.