Reciprocal and Inverse Trigonometric Functions
Reciprocal and Inverse Trigonometric Functions
A-Level Mathematics extends the basic trigonometric functions to include their reciprocals — secant, cosecant and cotangent — and their inverse functions. Understanding their definitions, graphs and properties is essential.
Definitions
The three reciprocal trigonometric functions are:
| Function | Definition | Written as |
|---|---|---|
| Secant | 1 / cosθ | secθ |
| Cosecant | 1 / sinθ | cosecθ |
| Cotangent | cosθ / sinθ = 1 / tanθ | cotθ |
These are not the same as the inverse trig functions. secθ means "one divided by cosθ", whereas arccos (or cos⁻¹) means "the angle whose cosine is...".
Graphs of Reciprocal Functions
y = secθ: This is the reciprocal of cosθ. It has:
- Vertical asymptotes wherever cosθ = 0 (at θ = 90°, 270°, etc.)
- A minimum value of 1 and maximum value of -1 (no values between -1 and 1)
- Period of 360° (or 2π radians)
- U-shaped curves pointing up where cosθ > 0 and pointing down where cosθ < 0
y = cosecθ: This is the reciprocal of sinθ. It has:
- Vertical asymptotes wherever sinθ = 0 (at θ = 0°, 180°, 360°, etc.)
- Same range restriction: |cosecθ| ≥ 1
- Period of 360°
y = cotθ: This is the reciprocal of tanθ. It has:
- Vertical asymptotes wherever sinθ = 0
- Period of 180° (or π radians)
- A decreasing function within each period
Key Identities Involving Reciprocal Functions
Two derived Pythagorean identities are particularly important:
- 1 + tan²θ = sec²θ (dividing sin²θ + cos²θ = 1 by cos²θ)
- 1 + cot²θ = cosec²θ (dividing sin²θ + cos²θ = 1 by sin²θ)
These are used extensively in integration (see Further Integration) and in solving equations.
Worked Example: Solve 2sec²θ = 5tanθ for 0 ≤ θ < 360°.
Step 1: Replace sec²θ using the identity: 2(1 + tan²θ) = 5tanθ
Step 2: Expand: 2 + 2tan²θ = 5tanθ
Step 3: Rearrange: 2tan²θ - 5tanθ + 2 = 0
Step 4: Factorise: (2tanθ - 1)(tanθ - 2) = 0
Step 5: tanθ = 1/2 gives θ = 26.57°, 206.57°; tanθ = 2 gives θ = 63.43°, 243.43°.
Inverse Trigonometric Functions
The inverse trigonometric functions arcsin, arccos and arctan (also written sin⁻¹, cos⁻¹, tan⁻¹) reverse the original functions. Because trig functions are many-to-one, their inverses are defined on restricted domains:
| Function | Domain | Range |
|---|---|---|
| arcsin(x) | -1 ≤ x ≤ 1 | -π/2 ≤ y ≤ π/2 |
| arccos(x) | -1 ≤ x ≤ 1 | 0 ≤ y ≤ π |
| arctan(x) | all real x | -π/2 < y < π/2 |
The graphs of inverse trig functions are reflections of the restricted original functions in the line y = x.
Key properties:
- arcsin and arctan are odd functions: arcsin(-x) = -arcsin(x)
- arccos is neither odd nor even, but arccos(-x) = π - arccos(x)
- arctan(x) has horizontal asymptotes at y = ±π/2
Differentiating Inverse Trig Functions
At A-Level, you should know:
- d/dx [arcsinx] = 1/√(1 - x²)
- d/dx [arccosx] = -1/√(1 - x²)
- d/dx [arctanx] = 1/(1 + x²)
These results follow from implicit differentiation. For example, if y = arcsinx then x = siny, so dx/dy = cosy = √(1 - sin²y) = √(1 - x²), and dy/dx = 1/√(1 - x²).
Worked Example: Using Identities
Show that cosecθ - sinθ = cosθcotθ.
Start from the left-hand side:
cosecθ - sinθ = 1/sinθ - sinθ
= (1 - sin²θ)/sinθ
= cos²θ/sinθ (using sin²θ + cos²θ = 1)
= cosθ × (cosθ/sinθ)
= cosθcotθ = right-hand side. QED.
Exam Tips
- When proving identities, work from one side only — do not manipulate both sides simultaneously.
- Sketch reciprocal graphs by first sketching the original (sin, cos, tan), marking zeros (which become asymptotes) and peaks/troughs (which become the turning points of the reciprocal).
- The identity 1 + tan²θ = sec²θ appears constantly in integration questions — recognise it instantly.
- When solving equations with mixed reciprocal functions, convert everything to sin and cos first, then simplify.