Unit 3: Trigonometric Functions

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).

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.

3. Fundamental Identities and Formulas

These identities are the building blocks for simplifying complex trigonometric expressions:

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