5.1 Trig Identities – Reciprocal and Quotient Identities

Pre-Calculus Tutorial

5.1 Trig Identities - Reciprocal and Quotient Identities


These identities deal with the reciprocal(s) of the problem given.

Sample Problem

If SIN(x) = 5/6, find CSC(x).


CSC and SIN are reciprocals of one another, and therefor, to find the answer, you just have to flip the values given for SIN(x)

SIN, COS, and TAN, all have reciprocals, I will include them below.

SIN (Sine) CSC (Cosecant)
COS (Cosine) SEC (Secant)
TAN (Tangent) COT (Cotangent)

How you can remember these effectively is no two start on the same letter. For example, the reciprocal of Sine, which starts with an “S”, is Cosecant, starting with a “C”. The same goes for Cosine “C” and Secant “S”, and Tangent “T” and Cotangent “C”. All six line up, and remain in the same spot, no matter the equation. So Cosecant will always be the reciprocal of Sine, no matter if the equation changes or not.

So, going back to the equation:

If SIN(x) = 5/6, Find CSC(x)

With the information above, we know that CSC, or Cosecant, is the reciprocal of SIN, or Sine, and therefor, we just need to flip the value given to SIN: 5/6

So, CSC(x) = 6/5.

Leave 6/5 like it is, don’t bother making it into a decimal or changing it in any way.

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