Just flip the familiar three: You already know sin, cos and tan. Their reciprocals (1 over each) get their own names:
secant sec θ = 1/cos θ
cosecant csc θ = 1/sin θ
cotangent cot θ = 1/tan θ = cos θ/sin θ.
A memory hook: look at the third letter. seCant pairs with cosine, coSecant pairs with sine. So sec goes with cos, csc goes with sin — the opposite of what you'd guess.
Where they blow up: A reciprocal is undefined wherever its denominator is 0.
sec θ is undefined when cos θ = 0 (at θ = 90°, 270°, …).
csc θ is undefined when sin θ = 0 (at θ = 0°, 180°, …).
cot θ is undefined when sin θ = 0 too (since cot = cos/sin).
IB-style question — exact reciprocal values
Find the exact value of sec(π/3), csc(π/6) and cot(π/4).
Step by step
- Write each as 1 over the basic ratio, then use exact values you know.
- Cosecant is 1 over sine.
- Cotangent is 1 over tangent.
Final answer
sec(π/3) = 2, csc(π/6) = 2, cot(π/4) = 1.
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Divide the old identity by cos² or by sin²: You already know sin²θ + cos²θ = 1. The two new identities come straight from it:
Divide every term by cos²θ → 1 + tan²θ = sec²θ.
Divide every term by sin²θ → cot²θ + 1 = csc²θ.
So you never have to memorise them cold — just remember which one to divide by. The identity with sec (cos's reciprocal) comes from dividing by cos²; the one with csc comes from dividing by sin².
IB-style question — find an exact value in a quadrant
θ is an angle with csc θ = 13/12 and θ lies in the first quadrant.
Find the exact value of cot θ.
Step by step
- Use 1 + cot²θ = csc²θ to link cot and csc directly.
- Subtract 1 (write 1 as 144/144).
- Take the square root. In the first quadrant every ratio is positive, so take the + root.
Final answer
cot θ = 5/12 (positive, because θ is in the first quadrant).
The sign comes from the quadrant: Taking a square root gives ±. The quadrant tells you which sign to keep.
Q1: all positive. Q2: only sin/csc positive. Q3: only tan/cot positive. Q4: only cos/sec positive.
Always read the given range before choosing the sign.