Algebra is Transformations
- Simplifying,
- Rearranging terms,
- Redrawing diagrams to look familiar,
- Applying known identities or equivalences to change terms into more recognizable forms.
Trig is Algebra on a Circle
The first transformation will be turning a problem into an ACUTE RIGHT TRIANGLE problem.
We learned about right triangles and the basics of Trigonometry way back in our Geometry classes. We were exposed to 3 of the 4 concepts required to survive and enjoy Trig:
- Pythagorean Theorem to determine the 3rd side of a right triangle,
- SOH-CAH-TOA to relate the length of the sides of a right triangle to the 3 basic trig functions of an acute angle,
- Special Right Triangles (30:60:90 and 45:45:90) for which we know the measures of the angles and the ratios of the side lengths.
But, of course, a little review might help.
- SOH-CAH-TOA allows us to determine the values of the trig functions of the ACUTE angles.
- Special Angles links together several concepts, including your textbook pg 20 diagram, solving/estimating quickly based on known values, and graphing (coming up in Ch 2).
Now, we spice things up a bit by moving out of Quadrant I. This is where the 4th knowledge requirement for Trig enters in: "A Smart Trig Class" or any other acronym to help us remember in which quadrants each of the trig function values will be positive or negative (ASTC).
- Reference Angles allow us to transform any angle back into an ACUTE angle.
- Reference Triangles allow us to transform problems back into ACUTE RIGHT TRIANGLE problems that we already know how to solve. Be sure to take notice of the arithmetic signs (+ -).
To get a graphical view of how the 6 trig functions fit together, check out:
- Trig Review from Fr Mike May. The lengths of the colored segments represent the trig function values of the angle made as you move the point around the circle.
- Dave's Short Course provides another, slightly different, view.