Reciprocal and Quotient Identities

Trigonometry · Axiom Academy

LESSON Reciprocal and Quotient Identities Understanding the fundamental relationships between trigonometric functions through the unit circle 1. The Three Reciprocal Identities Each of the three "co-functions" (cosecant, secant, cotangent) is defined as the reciprocal of one of the basic functions: 2. Visual Proof Using the Unit Circle The unit circle provides a geometric understanding of these identities. For any angle θ, we can see how the coordinates (cos θ, sin θ) relate to all six trigonometric functions. sin θ = y-coordinate = opposite/hypotenuse = y/1 cos θ = x-coordinate = adjacent/hypotenuse = x/1 The hypotenuse from origin to point has length 1 Reciprocals flip these relationships 3. The Two Quotient Identities Tangent and cotangent can be expressed as ratios (quotients) of sine and cosine. These follow naturally from the definitions on the unit circle: 4. Connecting All the Identities All five identities work together to show the deep connections between the six trigonometric functions. Here's the complete picture:

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