This question tests your understanding of the domain, range, and properties (even/odd, periodicity) of trigonometric functions.
| Function | Domain | Range |
|---|
| y=sinx | R (all real numbers) | [−1, 1] |
| y=cosx | R (all real numbers) | [−1, 1] |
| y=tanx | R∖{(2n+1)2π, n∈Z} | R |
| y=cotx | R∖{nπ, n∈Z} | R |
| y=secx | R∖{(2n+1)2π, n∈Z} | (−∞,−1]∪[1,∞) |
| y=cscx | R∖{nπ, n∈Z} | (−∞,−1]∪[1,∞) |
A function f is even if f(−x)=f(x) for all x in its domain (symmetric about the y-axis).
A function f is odd if f(−x)=−f(x) for all x in its domain (symmetric about the origin).
| Function | Even / Odd | Reason |
|---|
| cos(−x)=cosx | Even | Symmetric about y-axis |
| sec(−x)=secx | Even | Derived from cos |
| sin(−x)=−sinx | Odd | Anti-symmetric about origin |
| tan(−x)=−tanx | Odd | sin/cos ratio |
| cot(−x)=−cotx | Odd | cos/sin ratio |
| csc(−x)=−cscx | Odd | Derived from sin |
A function f is periodic with period T if f(x+T)=f(x) for all x, and T is the smallest such positive number.
| Function | Period |
|---|
| sinx | 2π |
| cosx | 2π |
| tanx | π |
| cotx | π |
| secx | 2π |
| cscx | 2π |
For any trigonometric function asked in Q5:
- State the domain — identify values of x where the function is undefined (division by zero).
- State the range — identify all possible output values.
- Classify as even or odd — test f(−x) and compare with f(x) and −f(x).
- State the period — the smallest T>0 such that f(x+T)=f(x).
Example: For f(x)=tanx:
- Domain: x=(2n+1)2π, n∈Z
- Range: R
- Odd function: tan(−x)=−tanx
- Period: π