Step by Step
CHO
Cosecant = Hyp/Opp
csc(θ) = 1/sin(θ) = Hypotenuse/Opposite — the reciprocal of sine.
SHA
Secant = Hyp/Adj
sec(θ) = 1/cos(θ) = Hypotenuse/Adjacent — the reciprocal of cosine.
CAO
Cotangent = Adj/Opp
cot(θ) = 1/tan(θ) = Adjacent/Opposite — the reciprocal of tangent.
Key
The "co-" prefix pairing rule
The "co-" prefix always pairs a function with a DIFFERENT base function, not the one it sounds like: Cosecant pairs with Sine (not cosine), and Secant pairs with Cosine (not sine). Cotangent pairs with tangent, which is the one exception that matches intuitively.
Applied Walkthrough
1
Given sin(θ) = 0.6, find csc(θ). Since cosecant is the reciprocal of sine (not cosine, despite starting with "co-"), csc(θ) = 1/sin(θ) = 1/0.6 ≈ 1.667.
2
Given cos(θ) = 0.8, find sec(θ). Since secant is the reciprocal of cosine (not sine, despite the name resembling "sine"), sec(θ) = 1/cos(θ) = 1/0.8 = 1.25.
3
This is the exact trap many students fall into: assuming cosecant relates to cosine, or secant relates to sine, simply based on how the names sound.
4
The safest way to avoid the trap: always remember the "co-" prefix goes with the OTHER function (cosecant↔sine, secant↔cosine), while cotangent↔tangent is the one pairing that does match its name.
Exam Application
Exams test specifically whether you know that cosecant is the reciprocal of sine (not cosine) and secant is the reciprocal of cosine (not sine) — a frequently reversed pairing due to how the names sound.
⚠ Common Trap
The single most common trap in this entire lesson: assuming cosecant is the reciprocal of cosine, or secant is the reciprocal of sine, based purely on how similar the names sound. It's the opposite — cosecant pairs with sine, secant pairs with cosine.
✓ Quick Self-Check
1. What is cosecant the reciprocal of?
Sine — csc(θ) = 1/sin(θ).
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2. What is secant the reciprocal of?
Cosine — sec(θ) = 1/cos(θ).
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3. What is cotangent the reciprocal of?
Tangent — cot(θ) = 1/tan(θ).
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4. If sin(θ) = 0.6, what is csc(θ)?
Approximately 1.667.
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5. What is the key rule for remembering which "co-" function pairs with which base function?
The "co-" prefix pairs with the OTHER function: cosecant↔sine, secant↔cosine (cotangent↔tangent is the one exception that matches its name).
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