Why do we care so much about triangles? Why don’t we have something like “trigonometric functions” for other shapes, like rectangles, pentagons, etc?

Furthermore, why does trigonometry focus on right triangles? What about other kinds of triangles?


  1. The relationship between distance/direction and coordinate axes.
  2. Triangulation: any shape with straight sides can be divided into triangles.

Trigonometry in Right Triangles

Defn. Suppose $x$ is in the open interval $(0,90)$, i.e., for $0<x<90$. Take any right triangle $\triangle O C D$ so that its acute angle at the vertex $O$ has $x$ degrees and so that $O C$ is the hypotenuse. Then $\angle D$ is the right angle, as shown:

By definition,
$$\sin x =\frac{|C D|}{|C O|}\left(=\frac{\text { opposite side }}{\text { hypotenuse }}\right)$$
$$\cos x =\frac{|O D|}{|C O|}\left(=\frac{\text { adjacent side }}{\text { hypotenuse }}\right)$$

Why is cosine called co-sine? Look at complementary angles in a right triangle.

What about angles bigger than $90$ or less than $0$?

Trigonometry in the Unit Circle

Defn. Suppose $t$ is any real number, and let $P_t$ be the image of the point $(1,0)$ after a clockwise rotation about the origin of $t$ degrees. Then the $x$-coordinate of $P_t$ is $\sin t$ and the $y$-coordinate of $P_t$ is $\cos t$.

  • Is $P_t$ a point on the unit circle? How do we know?
  • How do we justify the move from right triangles to the unit circle? If $t$ is in the open interval $(0,90)$, does the “right triangle” definition of sine and cosine agree with the “unit circle” definition?

Expressions vs Equations, Equations vs Identities

What is the difference between an equation and an identity?

Definition. An equation in one variable $x$ is a question that asks, when two expressions $f(x)$ and $g(x)$ are given, whether there is a number $k$ so that $f(x)$ is equal to $g(x)$ when $x=k$, i.e., so that $f(k)=g(k)$.

Definition. An identity $f(x)=g(x)$ is an equation which is true for all values of the variable (sometimes we allow a limited number of exceptions, for example values of $x$ for which one or both sides of the equation is undefined).

To solve an equation means to obtain all the solutions of the equations.

  • This can be done by starting with the given equation and applying a series of algebra steps, each preserving equality, until the solution or solutions are obtained.

What does it mean when we make a series of algebra steps, each preserving equality, in solving an equation? Is every such step reversible? Examples.

To prove an identity, we must establish that it is true for any value of $x$.

  • This can be done by starting with one side and applying a series of algebra steps, each preserving the value of the expression, to obtain the other side.
  • It can also be done by starting with a known identity and applying a series of algebra steps, each preserving equality, to obtain the desired identity.
  • Finally, one can also be start with the given identity and apply a series of algebra steps, as long as each step is reversible

Example. Prove the identity
$$1-\cos x=\frac{\sin ^{2} x}{1+\cos x}$$