Unit 3: Trigonometric Functions
Contents
1. Angles and Their Measurement
An angle is a measure of rotation of a given ray about its initial point. Rotation can be positive (anticlockwise) or negative (clockwise).
Degree and Radian Measure
- Degree Measure: If a rotation from initial to terminal side is 1/360th of a revolution, the angle is 1 degree (1°).
- Radian Measure: The angle subtended at the center by an arc of length 1 unit in a unit circle is 1 radian (1ᶜ).
π Radians = 180 Degrees
Radian Measure = (π / 180) × Degree Measure
Degree Measure = (180 / π) × Radian Measure
2. Trigonometric Ratios and Signs
Trigonometric ratios are defined based on the coordinates of a point on a unit circle. Their signs depend on the quadrant in which the terminal side of the angle lies.
| Quadrant | Rule (ASTC) | Positive Ratios |
|---|---|---|
| I (0 - 90°) | All | All (sin, cos, tan, etc.) are positive. |
| II (90 - 180°) | School | sin and cosec are positive. |
| III (180 - 270°) | To | tan and cot are positive. |
| IV (270 - 360°) | College | cos and sec are positive. |
3. Fundamental Identities and Formulas
These identities are the building blocks for simplifying complex trigonometric expressions:
1 + tan²x = sec²x
1 + cot²x = cosec²x
Sum and Difference Formulas:
- sin(A ± B) = sinA cosB ± cosA sinB
- cos(A ± B) = cosA cosB ∓ sinA sinB
- tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
4. Graphs of Trigonometric Functions
Trigonometric functions are periodic, meaning their values repeat after a fixed interval (Period).
- y = sin x: Period = 2π, Range = [-1, 1]. Passes through origin.
- y = cos x: Period = 2π, Range = [-1, 1]. Starts at (0, 1).
- y = tan x: Period = π, Range = R (all real numbers). Discontinuous at odd multiples of π/2.
5. General Solutions of Equations
When solving sin x = k, cos x = k, or tan x = k, we find the general solution that accounts for the periodic nature of the function.
- If sin x = sin α, then x = nπ + (-1)ⁿα
- If cos x = cos α, then x = 2nπ ± α
- If tan x = tan α, then x = nπ + α
6. Exam Focus Enhancements
- The ASTC Mnemonic: "After School To College" helps you remember which functions are positive in which quadrant.
- π Value: In degree-radian conversions, always keep π as a symbol unless specifically asked to use 22/7 or 3.14.
- Graph Symmetry: Remember that sin(-x) = -sin x (Odd), while cos(-x) = cos x (Even).
- Degree/Radian Confusion: Check your calculator mode before solving.
sin(30)in radians is very different fromsin(30°). - Signs in Quadrants: Forgetting that
tanis negative in the 2nd quadrant is a frequent error. - Undefined Tan: Be careful when x is 90° or 270°;
tan xbecomes undefined (Infinity).
Q: What is the relation between arc length, radius, and angle?
A: θ = l / r, where θ is the angle in radians, l is arc length, and r is the radius.
Q: What is the minimum and maximum value of (sin x + cos x)?
A: The values range from -√2 to +√2.