Unit 5 · Topic 3.11
The Secant, Cosecant, and Cotangent Functions
Build three reciprocal trigonometric functions from cosine, sine, and tangent, then use those relationships to explain their graphs and equations.
Learning Goals
- Define secant, cosecant, and cotangent using reciprocal and quotient relationships.
- Evaluate exact values with unit-circle information.
- Determine domains, ranges, periods, zeros, asymptotes, signs, and symmetry.
- Sketch reciprocal graphs from sine, cosine, and tangent graphs.
- Analyze transformed functions and solve reciprocal trigonometric equations.
1. Reciprocal Is Not Inverse
These are reciprocal functions. They are different from the inverse functions \(\arccos x\), \(\arcsin x\), and \(\arctan x\), which return principal angles.
2. Equivalent Quotient Forms and Restrictions
| Function | Equivalent form | Required restriction |
|---|---|---|
| \(\sec x\) | \(1/\cos x\) | \(\cos x\ne0\) |
| \(\csc x\) | \(1/\sin x\) | \(\sin x\ne0\) |
| \(\cot x\) | \(\cos x/\sin x=1/\tan x\) | \(\sin x\ne0\) |
The quotient \(\cos x/\sin x\) gives the full domain of cotangent directly. Writing only \(1/\tan x\) can hide the fact that tangent itself is undefined when \(\cos x=0\), even though cotangent equals 0 there.
3. Evaluate Exact Values
| Angle | Known base value | Reciprocal value |
|---|---|---|
| \(\pi/3\) | \(\cos(\pi/3)=1/2\) | \(\sec(\pi/3)=2\) |
| \(\pi/6\) | \(\sin(\pi/6)=1/2\) | \(\csc(\pi/6)=2\) |
| \(3\pi/4\) | \(\tan(3\pi/4)=-1\) | \(\cot(3\pi/4)=-1\) |
| \(2\pi/3\) | \(\cos(2\pi/3)=-1/2\) | \(\sec(2\pi/3)=-2\) |
Retain the sign when taking a reciprocal. A negative sine, cosine, or tangent produces a negative reciprocal value.
4. Compare Period and Symmetry
| Function | Period | Symmetry |
|---|---|---|
| \(\sec x\) | \(2\pi\) | Even |
| \(\csc x\) | \(2\pi\) | Odd |
| \(\cot x\) | \(\pi\) | Odd |
Taking a reciprocal preserves the period and even-or-odd behavior of the corresponding base function wherever both sides are defined.
5. Read Reciprocal Branches and Asymptotes
Secant and cosecant form separate upward- or downward-opening branches. Their vertical asymptotes occur exactly where the corresponding cosine or sine denominator is zero.
6. Build Secant from Cosine
For \(y=\sec x=1/\cos x\):
- Vertical asymptotes: \(x=\pi/2+k\pi\)
- Domain: all real \(x\) except those asymptote inputs
- Range: \((-\infty,-1]\cup[1,\infty)\)
- Period: \(2\pi\)
- Branch vertices: \((k\pi,(-1)^k)\)
Where cosine is positive, secant lies at or above 1. Where cosine is negative, secant lies at or below -1.
7. Build Cosecant from Sine
For \(y=\csc x=1/\sin x\):
- Vertical asymptotes: \(x=k\pi\)
- Domain: \(x\ne k\pi\)
- Range: \((-\infty,-1]\cup[1,\infty)\)
- Period: \(2\pi\)
- Vertices occur where \(\sin x=\pm1\), at \(x=\pi/2+k\pi\)
Cosecant and sine always have the same sign. As sine approaches 0, the magnitude of its reciprocal grows without bound.
8. Why Secant and Cosecant Have a Gap
Because \(|\sin x|\le1\) and \(|\cos x|\le1\), every defined reciprocal satisfies
Therefore neither function has outputs strictly between -1 and 1. The values \(\pm1\) are included because sine and cosine can equal \(\pm1\).
9. Build Cotangent from Sine and Cosine
- Vertical asymptotes: \(x=k\pi\), where sine is zero
- Zeros: \(x=\pi/2+k\pi\), where cosine is zero
- Range: all real numbers
- Period: \(\pi\)
- Behavior: decreasing from \(+\infty\) to \(-\infty\) between consecutive asymptotes
Cotangent's zeros occur exactly where tangent has vertical asymptotes; cotangent remains defined there because \(\sin x\ne0\).
10. Sketch from the Base Graph
- Graph the related sine, cosine, or tangent function lightly.
- Mark every zero of the denominator as a vertical asymptote.
- For secant or cosecant, mark reciprocal values where the base function equals \(\pm1\).
- Draw branches with the same sign as the base function.
- For cotangent, mark one zero halfway between consecutive asymptotes and draw a decreasing branch.
- Repeat using the correct period.
11. Transform a Secant Function
Analyze
- Period: \(2\pi/3\)
- Phase shift: right \(\pi/6\)
- Vertical shift: down 1
- Asymptotes: \(x=\pi/3+k\pi/3\)
- Range: \((-\infty,-3]\cup[1,\infty)\)
The vertices occur at \(x=\pi/6+k\pi/3\) and alternate between outputs 1 and -3.
12. Transform a Cotangent Function
Analyze
- Period: \(\pi/2\)
- Vertical asymptotes: \(x=k\pi/2\)
- Center-line crossings: \((\pi/4+k\pi/2,1)\)
- Range: all real numbers
Parent cotangent decreases between asymptotes. The negative outside coefficient reflects each branch, so this transformed function increases.
13. Solve Reciprocal Equations through Base Functions
Solve \(\sec x=-2\) on \(0\le x<2\pi\):
Similarly, \(\cot x=1\) is equivalent to \(\tan x=1\) where cotangent is defined, giving \(x=\pi/4+k\pi\).
14. Evaluate from Coordinates
If a terminal ray passes through \((-5,12)\), then \(r=13\), so
The Quadrant II coordinate signs provide a quick check on all three reciprocal values.
15. AP Workflow and Common Errors
- Identify the base function and denominator.
- Locate denominator zeros before drawing branches.
- State domain, range, period, signs, and symmetry.
- Use base-function extrema or zeros as reciprocal landmarks.
- Apply transformations to the landmarks and asymptotes.
- Do not confuse reciprocal functions with inverse trigonometric functions.
- Do not draw secant or cosecant through an asymptote.
- Do not give secant or cosecant outputs between -1 and 1.
- Do not place cotangent asymptotes where cosine is zero.
- Do not give cotangent the \(2\pi\) period of sine and cosine.
- Do not call \(|A|\) an amplitude for an unbounded reciprocal graph.
Key Takeaways
- Use reciprocal relationships to evaluate and graph secant, cosecant, and cotangent.
- Core relationship: \(\sec x=\frac{1}{\cos x},\quad\csc x=\frac{1}{\sin x},\quad\cot x=\frac{1}{\tan x}\)
- Error check: Do not confuse reciprocal functions with inverse trigonometric functions.
- State the domain, range, period, and asymptotes of \(y=\sec x\).
- Explain why \(y=\csc x\) has asymptotes where \(y=\sin x\) has zeros.
- List the asymptotes and zeros of \(y=\cot x\) on \(0\le x\le2\pi\).
- Analyze \(y=-2\csc(3(x-\pi/4))+4\).
- Solve \(\csc x=-\sqrt2\) on \(0\le x<2\pi\).