Topic: The Six Trigonometric Functions

Subtopic: Triangles

**Overview**

As you may suspect "trigonometry" is strongly tied to the study of "triangles", their measurement, and their relationships. In this section we develop the main theorems regarding triangles. Watch which involve right triangles only and which are more general.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 1.2.1 Solve problems using the "180° in a triangle" theorem
- 1.2.2 Solve problems using the Pythagorean theorem
- 1.2.3 Recognize and use the two "special" triangles (45°-45°-90° and 30°-60°-90°)
- 1.2.4 Recognize and use Pythagorean triples

**Terminology**

Terms you should be able to define: right triangle, hypotenuse, (triangle's) leg, acute triangle, obtuse triangle, equilateral triangle, equiangular triangle, isosceles triangle, theorem

Memorize the Pythagorean Theorem and know that while "a" and "b" are interchangable, "c" always represents the hypotenuse: `a^2+b^2=c^2`

Memorize the relationship of the sides on the two "special" triangles:

**Text Notes**

The e-textbook may cover several definitions and theorems from geometry (eg, the alternate interior angle theorem). These should be review from a high school geometry course. Skim as needed.

**Mini-Lectures and Examples**

STUDY: Angles, Triangles, and the xyr Definition of Trig Functions

**Supplementary Resources (recommended)**

The following links are for* *exploration. No need to explore them all, but pick at least two that interest you. Information on Pythagoras, the Pythagorean Theorem, and Pythagorean Triples:

- Biography of Pythagoras of Samos Pythagoreas lived from about 575 BC to about 495 BC. Pythagoras was a Greek philosopher who made important developments in mathematics, astronomy, and the theory of music. The theorem now known as the Pythagorean Theorem was known to the Babylonians 1000 years earlier, but he may have been the first to prove it.
- Proof of the Pythagorean Theorem (Cut the Knot has 121 more proofs if you are particularly interested in proofs :)
- History of the Pythagorean Theorem
- Common Pythagorean Triples (scroll down the page for chart/list)
- Pythagorean Triples Calculator (applet from Cut-The-Knot)

The Pythagorean theorem can be seen in terms of areas. In reference to the image at the right, the area of the yellow square (at the bottom) is `a^2`, the area of the blue square (at the right) is `b^2`, and these two areas together equal the area of the green square (tilted on the left top), `c^2`, thus `a^2+b^2=c^2`. This holds no matter how big the original red triangle (in the middle) is as long as it is a right triangle. Neat, aye? | |

This little video shows this area relationship using water. I want one of these! www.youtube.com/watch?v=CAkMUdeB06o |

**Supplemental Resources (optional)**

Videos from James Sousa's MathIsPower4U:

Animation: The Sum of the Interior Angles of a Triangle

Mini-lesson: Angle Relationships and Types of Triangles

rev. 2020-09-12