The Reciprocal Trig Ratios. Meet cosecant, secant, and cotangent — the three reciprocal trig ratios. Learn their definitions in terms of sides and coordinates, a memory trick for pairing them with their primary partners, and how to convert between reciprocal and primary ratios with worked examples.
The Unit Circle. Discover why a circle of radius 1 makes trig so elegant: every point on it is exactly (cos θ, sin θ). See how the special triangles scale down to fit inside the unit circle, and read off exact trig values directly from the coordinates of the 16 key points.
Exact Values Using the Unit Circle. Put the unit circle to work. Compare the unit-circle method against the CAST-plus-reference-angle method for finding exact values of all six trig ratios, including reciprocal ratios and negative angles.
Solving Trig Equations with All Six Ratios. Master the full strategy for solving equations that involve csc, sec, or cot: convert to the primary ratio first, find the reference angle, apply CAST, list principal angles, and apply domain restrictions. Five detailed examples including a decimal reciprocal case.
Inside the lesson — a free preview
What Is a Reciprocal Ratio?
Reciprocal Review. The reciprocal of a number is 1 divided by that number.
For example: the reciprocal of 5 is (1)/(5), and the reciprocal of (2)/(3) is (3)/(2).
The same idea applies to trig ratios — each of the three primary ratios has a reciprocal partner.
Pronunciation Guide. The three reciprocal ratios are pronounced:
= "co-secant"
= "secant"
= "co-tangent"
Memory Tip: The Third-Letter Rule
The Third-Letter Rule. The third letter of each reciprocal name tells you which primary ratio it pairs with:
c-s-c has "s" in the middle → reciprocal of sine
s-e-c has "c" → reciprocal of cosine
c-o-t has "t" → reciprocal of tangent
So: sine rightarrow cosecant, cosine rightarrow secant, tangent rightarrow cotangent.
Worked example: Express csc, sec, cot in Terms of x, y, and r
Example 1 — Reciprocal Ratios in Coordinate Form. Write each of the reciprocal trig ratios in terms of x, y, and r.
Recall the primary ratios: θ = (y)/(r), θ = (x)/(r), θ = (y)/(x). To find each reciprocal, flip the fraction.
Cosecant:
θ = (1)/(θ) = (1)/( y/r ) = (r)/(y)
Secant:
θ = (1)/(θ) = (1)/( x/r ) = (r)/(x)
Cotangent:
θ = (1)/(θ) = (1)/( y/x ) = (x)/(y)
Answer: θ = (r)/(y), θ = (r)/(x), θ = (x)/(y)
Create a free Studyio account to take the full lesson with voice-over teaching, practice questions and instant feedback.