Figure 1-22.-Function of an obtuse angle.can see that the values of x, y, and r are the same for135° as they are for 45°, exceptthatthevalueofxisnegative.From this it follows that the functions of anyobtuse angle are the same as the functions of itssupplement, except that any function in which x appearshas the opposite sign.The sine of an angle is y/r. Since x does not appearin this function, it follows that sin A = sin (180° – A).The cosine of an angle is fir. Since x appears in thisfunction, it follows that cos A = - cos (180° – A).The tangent of an angle is y/x. Since x appears inthis function, it follows that tan A = – tan ( 180°- A).The importance of knowing this lies in the fact thatmany tables of trigonometric functions list the functionsonly for angles to a maximum of 90°. Many obliquetriangles, however, contain angles larger than 90°. Todetermine a function of an angle larger than 90° from atable that stops at 90°, you lookup the function of thesupplement of the angle. If the function is a sine, youuse it as is. If it is a cosine or tangent, you give it anegative sign.The relationships of the function of obtuse anglesare as follows:The above relationships apply only when angle Ais greater than 90° and less than 180°.FUNCTIONS OF ANGLES INA RIGHT TRIANGLEFor an acute angle in a right triangle, the length ofthe side opposite the angle corresponds to y and thelength of the side adjacent to the angle corresponds tox, while the length of the hypotenuse corresponds to r.Therefore, the functions of an acute angle in a righttriangle can be stated as follows:If you consider a 90° angle with respect to the“circle of unit radius” diagram, you will realize that fora 90° angle, x = 0, y = 1, and r (as always) equals 1.Since sine = y/r, it follows that the sine of 90° = 1. Sincecosine = X/r, it follows that the cosine of 90° = 0/1, or0. Since tangent= y/x, it follows that tan 90° = 1/0, orinfinity (00). From one standpoint, division by 0 is amathematical inpossibility, since it is impossible tostate how many zeros there are in anything. From thisstandpoint, tan 90° is simply impossible. From anotherstandpoint it can be said that there arc an “infinite”number of zeros in 1. From that standpoint, tan 90° canbe said to be infinity.In real life, the sides of a right triangle y, x, and r,or side opposite, side adjacent, and hypotenuse, aregiven other names according to the circumferences. Inconnection with a pitched roof rafter, for instance, y orside opposite is “total rise,” x or side adjacent is “totalrun,” and r or hypotenuse is “rafter length.” Inconnection with a ground slope, y or side opposite is“vertical rise,” x or side adjacent is “horizontaldistance,” and r or hypotenuse is “slope distance.”1-19

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