Inputs
Leave blank to skip normalization.
| # | Function | a | b | c | d | e | Actions |
|---|
Results
Formulas Used
We solve linear equations in reciprocal trig functions:
a·sec(bx+c)+d = e ⇒ cos(bx+c) = 1 / ((e−d)/a)
a·csc(bx+c)+d = e ⇒ sin(bx+c) = 1 / ((e−d)/a)
a·cot(bx+c)+d = e ⇒ tan(bx+c) = 1 / ((e−d)/a)
For sec and csc, real solutions require |(e−d)/a| ≥ 1. For cot, y=0 implies bx+c = π/2 + πk. General solutions are shown in your chosen unit.
How to Use
- Select Function (sec, csc, or cot).
- Enter coefficients a, b, c, d, e for the linear form a·f(bx+c)+d=e.
- Choose Angle Unit, define the window [x_min, x_max], and optional normalization base.
- Click Compute. See general solutions and principal solutions within the window, normalized if requested.
- Use Plot to visualize zeros and singularities. Export results as CSV or PDF.
Visualization
g(x)=a·f(bx+c)+d−eThe plot shows g(x) across your interval. Zeros mark solutions. Dashed guides appear near singularities.
Example Data
| # | Function | a | b | c | d | e | Comment |
|---|---|---|---|---|---|---|---|
| 1 | sec | 2 | 1 | 0 | 0 | 3 | sec(x)=1.5 ⇒ cos(x)=2/3 |
| 2 | csc | 1 | 2 | 0 | -1 | 1 | csc(2x)-1=1 ⇒ sin(2x)=1 |
| 3 | cot | -1 | 1 | 0 | 0 | 2 | -cot(x)=2 ⇒ tan(x)=-1/2 |
| 4 | sec | 1 | 3 | 0.5 | 1 | 2 | sec(3x+0.5)+1=2 ⇒ cos(3x+0.5)=1 |
Domains, Ranges & Periods
Quick reference for reciprocal trig functions. θ is the inner angle bx+c. Period for x equals the θ-period divided by |b|.
| Function | Reciprocal of | Domain exclusions (θ) | Range | θ-period | x-period |
|---|---|---|---|---|---|
| sec θ | cos θ | θ ≠ π/2 + πk (±90° + 180°k) | (−∞, −1] ∪ [1, ∞) | 2π (360°) | 2π / |b| |
| csc θ | sin θ | θ ≠ πk (… , −180°, 0°, 180°, …) | (−∞, −1] ∪ [1, ∞) | 2π (360°) | 2π / |b| |
| cot θ | tan θ | θ ≠ πk | ℝ | π (180°) | π / |b| |
Reference Values (Common Angles)
Exact forms shown when simple; numeric approximations given to four decimals. “—” denotes undefined.
| θ (deg) | θ (rad) | sec θ | csc θ | cot θ |
|---|---|---|---|---|
| 0° | 0 | 1 | — | — |
| 30° | π/6 | 2/√3 ≈ 1.1547 | 2 | √3 ≈ 1.7321 |
| 45° | π/4 | √2 ≈ 1.4142 | √2 ≈ 1.4142 | 1 |
| 60° | π/3 | 2 | 2/√3 ≈ 1.1547 | 1/√3 ≈ 0.5774 |
| 90° | π/2 | — | 1 | 0 |
| 120° | 2π/3 | −2 | 2/√3 ≈ 1.1547 | −1/√3 ≈ −0.5774 |
| 135° | 3π/4 | −√2 ≈ −1.4142 | √2 ≈ 1.4142 | −1 |
| 150° | 5π/6 | −2/√3 ≈ −1.1547 | 2 | −√3 ≈ −1.7321 |
| 180° | π | −1 | — | — |
FAQs
An equation linear in sec, csc, or cot of a linear angle: a·f(bx+c)+d=e. It’s “linear” in the trig function, not the angle.
Because sec θ and csc θ are reciprocals of cos θ and sin θ. Since |sin θ|≤1 and |cos θ|≤1, their reciprocals satisfy |sec θ|≥1 and |csc θ|≥1.
It enumerates the general-solution families and filters x values that lie inside your chosen interval, converting between degrees, radians, or gradians as required.
The tool computes where sec or csc have cos or sin zeros, and where cot has sin zeros, then maps those θ values back to x within your interval.
Yes. Export PDF or CSV, and save your rows and settings to JSON or browser storage for quick reload later.