FUNCTIONS OF ANGLES IN A RIGHT TRIANGLE

Figure 1-22.-Function of an obtuse angle. can see that the values of x, y, and r are the same for 135° as they are for 45°, except that the value of x is negative. From this it follows that the functions of any obtuse angle are the same as the functions of its supplement, except that any function in which x appears has the opposite sign. The sine of an angle is y/r. Since x does not appear in this function, it follows that sin A = sin (180° – A). The cosine of an angle is fir. Since x appears in this function, it follows that cos A = - cos (180° – A). The tangent of an angle is y/x. Since x appears in this function, it follows that tan A = – tan ( 180°- A). The importance of knowing this lies in the fact that many tables of trigonometric functions list the functions only for angles to a maximum of 90°. Many oblique triangles, however, contain angles larger than 90°. To determine a function of an angle larger than 90° from a table that stops at 90°, you lookup the function of the supplement of the angle. If the function is a sine, you use it as is. If it is a cosine or tangent, you give it a negative sign. The relationships of the function of obtuse angles are as follows: The above relationships apply only when angle A is greater than 90° and less than 180°. FUNCTIONS OF ANGLES IN A RIGHT TRIANGLE For an acute angle in a right triangle, the length of the side opposite the angle corresponds to y and the length of the side adjacent to the angle corresponds to x, while the length of the hypotenuse corresponds to r. Therefore, the functions of an acute angle in a right triangle can be stated as follows: If you consider a 90° angle with respect to the “circle of unit radius” diagram, you will realize that for a 90° angle, x = 0, y = 1, and r (as always) equals 1. Since sine = y/r, it follows that the sine of 90° = 1. Since cosine = X/r, it follows that the cosine of 90° = 0/1, or 0. Since tangent= y/x, it follows that tan 90° = 1/0, or infinity (00). From one standpoint, division by 0 is a mathematical inpossibility, since it is impossible to state how many zeros there are in anything. From this standpoint, tan 90° is simply impossible. From another standpoint it can be said that there arc an “infinite” number of zeros in 1. From that standpoint, tan 90° can be said to be infinity. In real life, the sides of a right triangle y, x, and r, or side opposite, side adjacent, and hypotenuse, are given other names according to the circumferences. In connection with a pitched roof rafter, for instance, y or side opposite is “total rise,” x or side adjacent is “total run,” and r or hypotenuse is “rafter length.” In connection with a ground slope, y or side opposite is “vertical rise,” x or side adjacent is “horizontal distance,” and r or hypotenuse is “slope distance.” 1-19