Overview
This lecture introduces angle measurement units (degrees and radians), explains their relationships and conversion, and covers the basic properties and definitions of trigonometric functions on the unit circle.
Angle Measurement Units
- Angles can be measured in degrees (°) and radians (rad).
- 1 degree (1°) = 60 minutes (60') and 1 minute = 60 seconds (60").
- Radian is defined as the angle where the arc length equals the radius of the circle (1 radian).
- The circumference of a circle is 2πr, so a full circle is 2π radians or 360°.
Degree-Radian Conversion
- 360° = 2π radians.
- 1° = π/180 radians.
- 1 radian = 180/π degrees.
- Usually, angle in radians is written without a unit indicator, while degrees use the ° sign.
Unit Circle & Angle Representation
- The unit circle is centered at O with radius 1.
- Angle OAOM (α) is defined with one side OA and another OM.
- Points on the unit circle can represent multiple angles differing by integer multiples of 2π.
Trigonometric Functions
- Sine (sin α) = vertical coordinate (y) of point M.
- Cosine (cos α) = horizontal coordinate (x) of point M.
- Tangent (tan α) = sin α / cos α, defined when cos α ≠ 0.
- Cotangent (cot α) = cos α / sin α, defined when sin α ≠ 0.
- For all α, sin α and cos α are defined and range from -1 to 1.
- sin(α + 2kπ) = sin α and cos(α + 2kπ) = cos α for any integer k.
- tan(α) is undefined at α = π/2 + kπ; cot(α) is undefined at α = kπ.
Notable Angles & Values
- Frequently used angles: π/6, π/4, π/3 and their corresponding sine, cosine, tangent values.
- 0 and π/2 are special cases where tangent or cotangent are undefined.
Calculator Usage
- To input degrees, set the calculator to degree (D) mode.
- For radians, switch the calculator to radian (R) mode.
- Double check unit settings before calculations for trigonometric values.
Key Terms & Definitions
- Degree (°) — Traditional unit for measuring angles; a full circle is 360°.
- Radian (rad) — SI unit for angle, defined by arc length equal to radius.
- Unit Circle — Circle with radius 1 centered at the origin, used for defining trig functions.
- Sine (sin α) — y-coordinate of point on unit circle at angle α.
- Cosine (cos α) — x-coordinate of point on unit circle at angle α.
- Tangent (tan α) — Ratio of sin α to cos α; undefined when cos α = 0.
- Cotangent (cot α) — Ratio of cos α to sin α; undefined when sin α = 0.
Action Items / Next Steps
- Complete exercises 1 to 28 on pages 10–12.
- Do 10 True/False questions on page 22.
- Review and memorize sine, cosine, tangent values for π/6, π/4, π/3, 0, and π/2.
- Practice switching between degree and radian modes on your calculator.