Overview
This lecture covers converting degrees to radians, working with the unit circle, identifying exact trigonometric values using special triangles, determining equivalent trigonometric expressions, and finding radian values for given points on the Cartesian plane.
Radian Measure & the Unit Circle
- The unit circle has a radius of 1 and is key for measuring angles in radians.
- Common angles: PI/6 (30°), PI/4 (45°), PI/3 (60°), and their positions on the unit circle.
- Radian measure: PI radians = 180°, 2PI radians = 360°.
Special Triangles & Exact Values
- The 45-45-90 triangle (isosceles): sides 1, 1, √2; angles PI/4.
- The 30-60-90 triangle (from equilateral): sides 1, √3, 2; angles PI/6, PI/3.
- Use these triangles to find exact values, not decimals.
Equivalent Trigonometric Expressions
- To find an equivalent expression, relate the given angle to its acute angle and identify the sign using the CAST rule.
- In the second quadrant, sine is positive; in the third, tangent is positive; in the fourth, cosine is positive.
- Example: cos(5PI/6) = -cos(PI/6) = -√3/2.
Quadrant Sign Rules (CAST)
- C (IV): Cosine positive
- A (I): All positive
- S (II): Sine positive
- T (III): Tangent positive
Working with Cartesian Points
- To find the radian value for a point (x, y), use tan(θ) = y/x.
- When given a point, the quadrant helps determine the correct sign and angle.
- For negative outputs from tan⁻¹, convert to a positive angle within 0 to 2PI by adding 2PI or using 2PI − |angle|.
Multiple Solutions & Domain Checks
- If domain is from 0 to 2PI and signs of x/y are ambiguous, expect two solutions: θ = PI − angle and θ = 2PI − angle.
- Always check your calculator is in radians mode for trigonometric calculations in this unit.
Key Terms & Definitions
- Unit Circle — A circle with radius 1 centered at the origin, used for trigonometric calculations.
- Radian — Angle measure where PI radians = 180°.
- Special Triangles — Triangles with angles 30-60-90° and 45-45-90°, useful for exact trig values.
- CAST Rule — Mnemonic to remember which trig functions are positive in each quadrant.
- Equivalent Expression — A trigonometric expression using related acute angles and correct signs.
Action Items / Next Steps
- Practice writing equivalent expressions for trigonometric functions with radians.
- Make sure your calculator is set to radians before solving problems.
- Complete textbook problems from 7f on finding radian measures for given points.