Right Triangle Model and the Pythagorean Theorem Help
Introduction to Right Triangle Model and the Pythagorean Theorem
We have just 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 Δ PQR 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 , 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, remember that angles measured counterclockwise are considered positive, while angles measured clockwise are considered negative. If we have ∠ PQR that measures 30° around a circle in Cartesian coordinates, then ∠ RQP measures –30°, whose direction is equivalent to an angle of 330°. The cosines of these two angles happen to be the same, but the sines differ.
Ratios of Sides
Consider a right triangle defined by points P, Q , and R , as shown in Fig. 12-9. Suppose that ∠ QPR is a right angle, so Δ PQR 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. 12-9, 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.
Fig. 12-9. The right-triangle model for defining trigonometric functions. All right triangles obey the theorem of Pythagoras. Illustration for Problems 12-5 and 12-6.
Sum of Angle Measures and Range of Angles
Sum of Angle Measures
The following fact can be useful in deducing the measures of angles in trigonometric calculations. It’s a simple theorem in geometry that you should remember. In any triangle, the sum of the measures of the interior angles is 180° (π rad). This is true whether it is a right triangle or not, as long as all the angles are measured in the plane defined by the three vertices of the triangle.
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