This topic extends trigonometry by introducing the reciprocal trigonometric ratios, their identities and the inverse trigonometric functions.
Understanding these requires familiarity with the unit circle and the basic trigonometric ratios sinθ, cosθ and tanθ.
| Concept | Meaning / Explanation |
|---|---|
| secθ | secθ = 1 ÷ cosθ
Undefined when cosθ = 0. |
| cosecθ | cosecθ = 1 ÷ sinθ
Undefined when sinθ = 0. |
| cotθ | cotθ = cosθ ÷ sinθ
Undefined when sinθ = 0. |
| Pythagorean identities (AHL) | 1 + tan2θ = sec2θ
1 + cot2θ = cosec2θ |
| arcsin(x) | Inverse of sinθ, giving the angle whose sine is x.
Domain: −1 ≤ x ≤ 1 Range: −π ÷ 2 ≤ arcsin(x) ≤ π ÷ 2 |
| arccos(x) | Inverse of cosθ, giving the angle whose cosine is x.
Domain: −1 ≤ x ≤ 1 Range: 0 ≤ arccos(x) ≤ π |
| arctan(x) | Inverse of tanθ.
Domain: all real x Range: −π ÷ 2 < arctan(x) < π ÷ 2 |
📌 1. Understanding reciprocal trig ratios
Reciprocal trig ratios arise naturally from the unit circle:
- If cosθ is the x-coordinate on the unit circle, then secθ = 1 ÷ cosθ measures how “large” the reciprocal of that horizontal distance is.
- If sinθ is the y-coordinate, then cosecθ = 1 ÷ sinθ measures the reciprocal vertical distance.
- cotθ = cosθ ÷ sinθ is the reciprocal of tanθ.
These values often appear in calculus, trigonometric simplifications and differential equations, especially in advanced mathematics.
📌 2. Pythagorean identities (AHL extensions)
Starting from the basic identity:
cos2θ + sin2θ = 1
Divide through by cos2θ:
tan2θ + 1 = sec2θ
Divide through by sin2θ:
1 + cot2θ = cosec2θ
These identities help simplify expressions and solve trigonometric equations in a wider range of contexts.
📌 3. Inverse trigonometric functions: definitions, domains & ranges
Each inverse trigonometric function “undoes” its original function, but because trigonometric functions are periodic, they are restricted to specific ranges to ensure they are functions.
- arcsin(x) returns angles only from −π ÷ 2 to π ÷ 2.
- arccos(x) returns angles only from 0 to π.
- arctan(x) returns angles from −π ÷ 2 to π ÷ 2, not including endpoints.
Knowing these domains and ranges is crucial when solving equations like sinθ = 0.4 or tanθ = 2.
Example — solving using inverse functions
Solve sinθ = 0.6 for θ in the principal range.
θ = arcsin(0.6)
arcsin(0.6) returns an angle between −π ÷ 2 and π ÷ 2.
Using GDC: θ ≈ 0.6435 radians.
🧮 GDC Use
- Use inverse trig keys to compute arcsin, arccos and arctan values accurately.
- Graph reciprocal functions (secx, cosecx, cotx) to identify asymptotes and discontinuities.
- Use graph intersection methods to solve equations like secx = 2 or tanx = 3, by plotting the graph of the function and using x calc to find the answer.
- Always ensure the calculator is in radian mode for AHL trigonometry unless stated otherwise.
🌍 Real-World Applications
- Inverse trig functions appear frequently in navigation, engineering angles and direction calculations.
- secθ, cosecθ and cotθ are used in advanced mathematics and physics, including wave analysis and differentiation of trig functions.
- Computer graphics use arctan functions to determine camera angles and object orientation.
🔍 TOK Perspective
- Inverse functions illustrate how we decide which parts of a periodic function to treat as “principal” — is this a discovery or a convention?
- Reciprocal trig ratios highlight how mathematical structures extend logically — does mathematics grow from internal necessity or external usefulness?
- Different cultures contributed to trigonometric ideas; does this influence how we perceive the “facts” of mathematics?
📝 Paper Tips
- Always check whether the inverse trig answer must be in radians or degrees.
- Know the domain and range of arcsin, arccos and arctan — many mistakes come from giving the wrong branch.
- When simplifying trig expressions, use reciprocal identities to reduce complexity.
- Sketch a unit circle to understand signs (positive/negative) in different quadrants.