**Introduction to the Right Triangle Model—Triangle and Angle Notation**

Trigonometry involves countless relationships among lines, angles, and distances. It seems that each situation has its own function or formula. Throw in the Greek symbology, and things can look scary. But all complicated structures are built using simple blocks, and difficult problems can be unraveled (or concocted, if you like) using circular trigonometric functions.

In the previous chapter, we defined the six circular functions—sine, cosine, tangent, cosecant, secant, and cotangent—in terms of points on a circle. There is another way to define these functions: the *right-triangle model.*

**Triangle And Angle Notation**

In geometry, it is customary to denote triangles by writing an uppercase Greek letter delta (Δ) followed by the names of the three points representing the corners, or *vertices,* of the triangle. For example, if *P* , *Q* , and *R* are the names of three points, then *ΔPQR* is the triangle formed by connecting these points with straight line segments. We read this as “triangle *PQR.* ”

Angles are denoted by writing the symbol ∠ (which resembles an extremely italicized, uppercase English letter L without serifs) followed by the names of three points that uniquely determine the angle. This scheme lets us specify the extent and position of the angle, and also the rotational sense in which it is expressed. For example, if there are three points *P* , *Q* , and *R,* then *∠PQR* (read “angle *PQR”* ) has the same measure as *∠RQP,* but in the opposite direction. The middle point, *Q* in either case, is the *vertex* of the angle.

The rotational sense in which an angle is measured can be significant in physics, astronomy, and engineering, and also when working in coordinate systems. In the Cartesian plane, angles measured counterclockwise are considered positive by convention, while angles measured clockwise are considered negative. If we have *∠PQR* that measures 30° around a circle in Cartesian coordinates, then *∠RQP* measures –30°, which is the equivalent of 330°. The cosines of these two angles happen to be the same, but the sines differ.

**Sides and Angles**

**Ratios Of Sides**

Consider a right triangle defined by points *P* , *Q* , and *R,* as shown in Fig. 2-1. Suppose that *∠QPR* is a right angle, so *ΔPQR* is a *right triangle.* Let *d* be the length of line segment *QP* , *e* be the length of line segment *PR,* and *f* be the length of line segment *QR.* Let *θ* be *∠PQR,* the angle measured counterclockwise between line segments *QP* and *QR.* The six circular trigonometric functions can be defined as ratios between the lengths of the sides, as follows:

sin *θ* = *e/f*

cos *θ* = *d/f*

tan *θ* = *e/d*

csc *θ* = *f/e*

sec *θ* = *f/d*

cot *θ* = *d/e*

The longest side of a right triangle is always opposite the 90° angle, and is called the *hypotenuse.* In Fig. 2-1, this is the side *QR* whose length is *f* . The other two sides are called *adjacent sides* because they are both adjacent to the right angle.

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