Fundamental Identities

Pre-Calculus · Axiom Academy

The reciprocal, quotient, and even–odd identities — the building blocks every other trig identity is made of. Three of the six functions are defined as the reciprocals of the other three. Whenever a value v moves, its reciprocal 1/v moves the opposite way — small numbers flip to large ones and back — yet their product is always exactly 1 . That fixed product is the whole identity. Each function pairs with one reciprocal partner Equivalently, every pair multiplies to 1 On the unit circle a point sits at . The line from the center to that point rises by and runs by , so its slope is — and that slope is exactly . Cotangent is the same ratio upside down. The height of the point above the horizontal axis. The horizontal distance from the center to under the point. — the steepness of the radius. — run over rise, the reciprocal slope. Slope is "change in y over change in x ." Going from the center (0,0) out to , the change in y is and the change in x is , so the slope is . The radius and the tangent value are the same number. Negating the angle reflects the point on the unit circle across the horizontal axis. The x -coordinate is unchanged but the y -coordinate flips sign. Since is the x -coordinate and is the y -coordinate, cosine stays the same (it is even ) while sine flips (it is odd ). Cosine and secant are even. They depend only on the unchanged x -coordinate, so and .

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