Quick Summary
Reciprocal Functions
sec x = 1/cos x, cosec x = 1/sin x, cot x = 1/tan x. They have vertical asymptotes and specific ranges.
Key Identities
tan²x + 1 = sec²x and 1 + cot²x = cosec²x. These come from dividing the Pythagorean identity.
Inverse Functions
arcsin, arccos, arctan are the inverses of sin, cos, tan. They must have restricted domains to be one-to-one.
Domains & Ranges
Inverse functions have limited input and output values. Reciprocal functions are undefined where their originals equal zero.
Definitions of Reciprocal Trigonometric Functions
What Are Reciprocal Trigonometric Functions?
Think of reciprocal functions as the "flipped fraction" versions of the main trigonometric functions. If sin x gives you a ratio, then cosec x gives you the inverse of that ratio. They're not inverses in the "undo the operation" sense — they're reciprocals in the algebraic sense (the fraction is flipped).
There are three reciprocal trigonometric functions, one for each main trig function:
Understanding the Definition
Secant (sec) is the reciprocal of cosine. Wherever cosine is small, secant is large. Wherever cosine is zero, secant is undefined (vertical asymptote). Think of it like this: if cos x = 0.5, then sec x = 1/0.5 = 2.
Cosecant (cosec) is the reciprocal of sine. Same logic applies. When sin x gets close to zero, cosec x shoots off to infinity.
Cotangent (cot) is the reciprocal of tangent. It can also be written as cos x / sin x — this is super useful because you can work with it using sine and cosine instead of trying to use the tan definition.
💡 Key Insight: When solving equations with reciprocal trig functions, convert them back into the regular trig functions first. For example, if you see sec x = 2, rewrite it as 1/cos x = 2, then cos x = 1/2. Much easier!
Where Are These Functions Undefined?
- sec x is undefined where cos x = 0, which is at odd multiples of 90° (90°, −90°, 270°, etc.) or odd multiples of π/2.
- cosec x is undefined where sin x = 0, which is at multiples of 180° (0°, 180°, 360°, etc.) or multiples of π.
- cot x is undefined where sin x = 0 (same as cosec), which is at multiples of 180° or π.
Practice Question
Find the exact value of sec 60°.
Graphs of Reciprocal Trigonometric Functions
The graphs of reciprocal trig functions are formed from the graphs of the regular trig functions. Where the regular function has a small value, the reciprocal gets large. Where it hits zero, the reciprocal shoots to infinity (vertical asymptote). Understanding this connection is absolutely crucial.
The Graph of y = sec x
Key features:
- Period: 360° or 2π radians (same as cos x)
- Vertical asymptotes: at odd multiples of 90° (−450°, −270°, −90°, 90°, 270°, 450°, etc.)
- Domain: all x except odd multiples of 90°
- Range:
- Shape:
The Graph of y = cosec x