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The Right Triangle Model Help

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

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.

A Flurry of Facts The Right Triangle Model Ratios Of Sides

Fig. 2-1 . The right-triangle model for defining trigonometric functions. All right triangles obey the theorem of Pythagoras.

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