Trigonometric Functions and the Unit Circle

1. Trigonometric Functions and the Unit Circle source: SPSU.edu

2. Lesson Plan 1. Unit circle 4. Trigonometric function #3 – tangent a. definition/features a. definition b. illustration b. examples c. implications 2. Trigonometric function #1 – sine a. definition 5. Inverse trigonometry b. memorization tools a. definition(s) c. examples b. calculations c. examples 3. Trigonometric function #2 – cosine a. definition 6. Worksheet b. memorization tools c. examples

3. The Unit Circle • definition – a circle of radius 1 centered at the origin (0,0) in the Cartesian coordinate system in the Euclidean plane (for the purposes of this lesson, angles will be written/calculated in degree mode (0° to 360°), not radian mode (0 to 2)). • created by graphs of sine, cosine functions (see illustration below) ; for an animated version of how the unit circle is created, click here. • used to simplify, summarize calculations related to trigonometric functions

4. Sine Function • definition – in a triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse, i.e. for the following triangle: • however, in terms of the unit circle, the sine value of an angle is its corresponding “y” value on the unit circle, i.e. for  = 30°: sin() =  y-coordinate of point where 30° and the unit circle intersect

5. Sine Function (continued) • memorization tools: 1. SOHcahtoa (Sine = Opposite over Hypotenuse) 2. “sine starts in the sand” (sin(0) = 0) 3. sine values are positive in quadrants I and II (since the y values are positive there) • examples: 1. sin(45°) = y-coordinate of point on unit circle indicated by 45° = (illustrated by the green line) 2. sin(270°) = y-coordinate of point on unit circle indicated by 270° = -1 (illustrated by the blue line) 3. given 9 14 2 2 9 sin() =  14

6. Cosine Function • definition – in a triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse, i.e. for the following triangle: • however, in terms of the unit circle, the cosine value of an angle is its corresponding “x” value on the unit circle, i.e. for  = 60°: cos() =  x-coordinate of point where 60° and the unit circle intersect

7. Cosine Function (continued) • memorization tools: 1. sohCAHtoa (Cosine = Adjacent over Hypotenuse) 2. “cosine starts in the sand” (cos(0) = 1) 3. cosine values are positive in quadrants I and IV (since the x values are positive there) • examples: 1. cos(45°) = x-coordinate of point on unit circle indicated by 45° = (illustrated by the orange line) 2. cos(180°) = x-coordinate of point on unit circle indicated by 180° = -1 (illustrated by the brown line) 3. given 11 16 2 2 16 cos() =  11

8. Tangent Function • definition – in a triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side, i.e. for the following triangle: • however, in terms of the unit circle, the tangent value of an angle is its “y” value on the unit circle divided by its “x” value , i.e. for  = 60°: tan() =  y-coordinate at 60° x-coordinate at 60°

9. Tangent Function (continued) • memorization tools: 1. sohcahTOA (Tangent = Opposite over Adjacent) 2. tangent values are positive in quadrants I and III, since x and y values are both positive in quadrant I and since x and y values are both negative in quadrant III • examples: 1. tan(45°) = x-coordinate of point on unit circle indicated by 45° divided by y-coordinate at 45° = ÷ = 1 2. tan(180°) = 0 ÷ 1 = 0 3. given 7 13 2 2 2 2 7 tan() =  13

10. Tangent Function (continued) • there are many implications of the tangent function for future lessons, including the application of differentiation, as illustrated below by tangential lines in 2-dimensional space and tangential planes in 3-dimensional space. tangent line (in green) tangent plane (in blue) source: wolfram.mathworld.com

11. Inverse Trigonometry • in mathematical concepts involving inverse trigonometry, instead of asking “what is the sine value on the unit circle at 180°?” one is asked “at what degree value on the unit circle is the sine value 0?” (in this instance the 2 questions refer to the same point on the unit circle, namely (-1,0)) • IMPORTANT NOTE – when asking for degree values on the unit circle whose sine, cosine and tangent values are specific, it is necessary to define the limit within which the degree values can exist; otherwise an infinite number of answers are possible, as the unit circle circumnavigates the origin (potentially) infinitely-many times (see illustration below) . Click here to perform an interactive activity which illustrates this notion. onwards to infinity… (if not bounded)

12. Inverse Trigonometry (continued) • inverse of sine  arcsine (denoted by “arcsin”) • inverse of cosine  arccosine (denoted by “arccos”) • inverse of tangent  arctangent (denoted by “arctan”) • examples: 1. between 0° and 360°, arcsin( ) = 45°, 135° 2. 0°  x  360°, arccos(- ) = 150°, 210° 2 2 3 2

13. Practice Now complete the accompanying worksheet using the Powerpoint presentation and the links included in it. Don’t hesitate to ask questions (by raising your hand) or work with other students around you. Good luck! If you finish the worksheet and want extra practice, quiz yourself by going here, removing certain features of the interactive unit circle and filling them back in.