Useful Notes on the 6 Trig Functions

by Mr. Bergen (bergen@chapin.edu)

prepared for students at Saddle River Day School Tuesday 3/6/07

Things You Should Know Prior to this Lesson

• The basic facts of a 45-45-90 right triangle, sometimes nicknamed an isosceles triangle
  • comes from drawing a diagonal in a square
  • the hypotenuse is square root of 2 times a side
  • both legs are equal
• The basic facts of a 30-60-90 right triangle, which you can get by splitting an equilateral triangle in half
  • the shorter leg (SL) is opposite the 30 degree angle
  • the longer leg (LL) is opposite the 60 degree angle
  • the longer leg (LL) is square root of 3 times the shorter leg (SL)
  • the hypotenuse (LL) is twice the shorter leg (SL)
  • Click here for info about the 45-45-90 triangle and the 30-60-90 triangle
• there are two ways to measure angles: degrees and radians ... you should know the common angles around a circle
• Click here for COMMON ANGLES around a circle with an applet for testing yourself
• there are 360 degrees in a circle
• there are 2PI radians in a circle, which means about 6.28 radian in a circle, so 1 radian is about 1/6th of a circle or about 57 degrees
• Click here for a wonderful explanation of WHAT IS A RADIAN?
• SOHCAHTOA gives us the SIN, COS and TAN in a funky one word expression

Overview

• Step 1: Let us find the sin, cos, tan of the 5 basic angles which are 0, 30, 45, 60, 90 in degrees or ??? in radians
• Step 2: let's think about the unit circle and see how these things relate to a point moving around the circle
• Step 3: let's "get creative" and figure out how to find the SIN, COS, TAN of angles beyond 90 degrees ... let us note that no longer does SOHCAHTOA make any sense but we have moved to a new understanding base on the unit circle

Some observations about my trig wheel

• Each end of a spoke are reciprocals, e.g. A = 1/B or A*B=1
• If A,B,C are any 3 in a circle, then A=B/C; this works regardless of how you go around the circle
• Click here for an excellent treatment of TRIG and RIGHT TRIANGLES
• Click here for the 6 TRIG functions

Some other useful links on this topic ... this one is really cool visual ... click

• Click here for a listing of TRIG IDENTITIES
• Click here for the defintion of the SINE FUNCTION as part of the unit circle and an applet that lets you test out the SIN different values from 360 to -360
• Click here for the defintion of the COSINE FUNCTION as part of the unit circle and an applet that lets you test out the COS of different values from 360 to -360
• Click here for a MORE COMPLICATED but interesting definition of the TANGENT FUNCTION with an applet for testing values
• Click here for a COSINE problem
• Click here for a SINE problem
• Click here for a TANGENT problem