Course: Trigonometry
Topic: The Six Trigonometric Functions
Subtopic: Triangles


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.


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


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:
Graphic of the two special triangles one with sides x, x, x times square root of 2; the other with sides x, x times square root of 3, 2 times x.

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:

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? Red right triangle with yellow square on the 'a' leg, blue square on the 'b' leg, big green square on the 'c' hypotenuse.
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