1. Understanding Signs of Trigonometric Functions
The sign of each trigonometric function depends on the quadrant in which the terminal side of the angle lies. This is based on the coordinate plane signs of x and y.
Use the ASTC rule:
- I Quadrant — All functions are positive
- II Quadrant — Sin and cosec positive
- III Quadrant — Tan and cot positive
- IV Quadrant — Cos and sec positive
2. Sign Table for All Six Trigonometric Functions
| Function | Quadrant I | Quadrant II | Quadrant III | Quadrant IV |
|---|---|---|---|---|
| sin | + | + | - | - |
| cos | + | - | - | + |
| tan | + | - | + | - |
| cot | + | - | + | - |
| sec | + | - | - | + |
| cosec | + | + | - | - |
3. Domain of Trigonometric Functions
The domain tells us the set of angle values for which each function is defined. Some functions have restrictions because division by zero is not allowed.
3.1. Domain Rules
- sin θ and cos θ → defined for all real θ
- tan θ → undefined when \(\cos θ = 0\), i.e.,
\( θ ≠ \dfrac{π}{2} + nπ \)
- cot θ → undefined when \(\sin θ = 0\), i.e.,
\( θ ≠ nπ \)
- sec θ → undefined when \(\cos θ = 0\)
- cosec θ → undefined when \(\sin θ = 0\)
4. Range of Trigonometric Functions
The range is the set of possible output values a trigonometric function can take.
4.1. Range Rules
- sin θ →
\( [-1, 1] \)
- cos θ →
\( [-1, 1] \)
- tan θ → all real numbers
- cot θ → all real numbers
- sec θ →
\( (-∞, -1] ∪ [1, ∞) \)
- cosec θ →
\( (-∞, -1] ∪ [1, ∞) \)
5. Summary Table: Domain and Range
| Function | Domain | Range |
|---|---|---|
| sin θ | All real numbers | [-1, 1] |
| cos θ | All real numbers | [-1, 1] |
| tan θ | All real except \(\dfrac{π}{2} + nπ\) | All real |
| cot θ | All real except \(nπ\) | All real |
| sec θ | All real except \(\dfrac{π}{2} + nπ\) | \((-∞, -1] ∪ [1, ∞)\) |
| cosec θ | All real except \(nπ\) | \((-∞, -1] ∪ [1, ∞)\) |
6. Examples for Better Understanding
Example 1: Why is tan 90° undefined?
Because:
\( \tan θ = \dfrac{\sin θ}{\cos θ} \)
\( \cos 90° = 0 \Rightarrow \dfrac{1}{0} \) is undefined.
Example 2: Can sin θ ever be 2?
No, because its range is only between –1 and 1.