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TRIGONOMETRY Our discussion will focus primarily on the study of plane trigonometry. It is intended only as a review of the relationships among the sides and angles of plane triangles and their ratios, called the TRIGONOMETRIC FUNCTIONS. The information presented here is based on Mathematics, Vol. 1, NAVEDTRA 10069-D1, Mathematics, Vol. 2-A, NAVEDTRA 10062, chapters 3, 4, and 6. Spherical trigonometry will be covered as you advance in rate. It is a prerequisite to the study of navigation, geodesy, and astronomy. Hence, the subject of spherical trigonometry will be introduced in the Engineering Aid class C1 school curriculum.When two straight lines intersect, an angle is formed, You can also generate an angle by rotating a line having a set direction, Figure 1-19 depicts the generation of an angle. The terminal line OB is generated from the initial point OA and forms L AOB, which we will call f (Greek letter, pronounced "theta"). Angle f is generally ex-pressed in degrees. The following paragraphs will discuss the degree and the radian systems that are generally used by Engineering Aids. The DEGREE SYSTEM is the most common system used in angular measurement. Angular measurement by REVOLUTION is perhaps the unit you are most familiar with. In the degree system, a complete revolution is divided into 360 equal parts called degrees (360). Each degree is divided into 60 minutes (60), and each minute into 60 seconds (60"). For convenience in trigonometric computations, the 360 is divided into four parts of 90 each. The
Figure 1-19.-Generation of an angle, resulting angle measured in degrees. 90 sectors, called QUADRANTS, are numbered counterclockwise starting at the upper right-hand sector. When the unit radius r (the line generating the angle) has traveled less than 90 from its starting point in a counterclockwise direction (or, as conventionally referred to as, in a positive direction), the angle is in the FIRST quadrant (I). When the unit radius lies between 90 and 180, the angle is in the SECOND quadrant (II). Angles between 180 and 270 are said to lie in the THIRD quadrant (III), and angles greater than 270 and less than 360 are in the FOURTH quadrant (IV).When the line generating the angle passes through more than 360, the quadrant in which the angle lies is found by subtracting from the angle the largest multiple of 360 that the angle contains and determining the quadrant in which the remainder falls.The RADIAN SYSTEM of measuring angles is even more fundamental than the degree system. It has certain advantages over the degree system, for it relates the length of arc generated to the size of the angle and the radius. The radian measure is shown in figure 1-20. If the length of the arc (s) described by the extremity of the line segment generating the angle is equal to the length of the line (r), then it is said that the angle described is exactly equal to one radian in size; that is, for one radian, s = r.The circumference of a circle is related to the radius by the formula, C = 2pr. This says that the circumference is 2p times the length of the radius. From the relationship of arc length, radius, and radians in the preceding paragraph, this could be extended to say that a circle
Figure 1-20.-Radian measure.
Figure 1-21.-Circle of unit radius with quadrants shown. contains 2pr radians, and the circumference encompasses 3600 of rotation. It follows that
By dividing both sides of the above equation by p, we find that
As in any other formula, you can always convert radians to degrees or vice versa by using the above relationship. |
 
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