Overview
This lecture explains how to find the exact value of trigonometric functions at special angles in radians, using the unit circle and symmetry.
The Unit Circle and Special Angles
- The unit circle represents angles in radians, with x and y values corresponding to cosine and sine, respectively.
- Key radian marks: 0, π/6, π/4, π/3, and π/2 in quadrant 1.
- Special angle values for cosine (x) move from 1, √3/2, √2/2, 1/2, to 0 as the angle increases from 0 to π/2.
- Special angle values for sine (y) move from 0, 1/2, √2/2, √3/2, to 1 as the angle increases from 0 to π/2.
Using Symmetry for Other Quadrants
- Special angle values can be found in other quadrants by considering symmetry about the unit circle.
- For quadrant 2, the sine value remains the same as quadrant 1, but the cosine (x value) becomes negative.
Example: Finding Cosine of 3π/4
- 3π/4 is a multiple of π/4 and lies in quadrant 2.
- The cosine of 3π/4 is the negative of the cosine of π/4: cos(3π/4) = -√2/2.
- The sine of 3π/4 is the same as the sine of π/4: sin(3π/4) = √2/2.
Key Terms & Definitions
- Unit Circle — A circle with radius 1 centered at the origin, used to define trigonometric functions.
- Special Angles — Common angles with well-known sine and cosine values (π/6, π/4, π/3, etc.).
- Radians — A unit of angle measure; π radians equals 180 degrees.
Action Items / Next Steps
- Practice finding sine and cosine values for other special angles in different quadrants.
- Review the values for all special angles on the unit circle.