Y B Ans. . 18. Trigonometric functions applied to right triangles. When the angle is acute, the abscissa, ordinate, and distance for any point in the terminal side, form a right triangle, in which the given angle is one of the acute angles. On account of the many applications of the right triangle in trigonometry, the definitions of the trigonometric functions will be stated with special reference to -X the right triangle. A b a Draw the right triangle ABC, Fig. 23, with the vertex A at the origin, and AC on the initial line. Then AC and CB are the coördinates of B in the terminal side AB. Let AC = b, FIG. 23. a, and AB = C. By definition: abscissa b distance c = a = = = side adjacent side opposite hypotenuse side adjacent hypotenuse side opposite Y Again, suppose the triangle ABC placed so that B has its vertex at the origin, BC for the initial side, and BA for the terminal side, as in Fig. 24. The coördinates of A are BC = a and CA 24 = b. B a FIG. 24. Then, no matter where the right triangle is found, the functions of the acute angles may be written in terms of the legs and the hypotenuse of the right triangle. 19. Relations between the functions of complementary angles. From the formulas of Art. 18, the following relations are evident: But angles A and B are complementary; therefore, the sine, cosine, tangent, cotangent, secant, and cosecant of an angle are respectively the cosine, sine, cotangent, tangent, cosecant, and secant of the complement of the angle. They are also called co-functions. For example, cos 75° = sin (90° — 75°) = sin 15°; tan 80° = cot (90° — 80°) = cot 10°. Note. The term cosine was not used until the beginning of the 17th century. Before that time the expression, sine of the complement (complementi sinus) was used instead. Cosine is a contraction of this. Similarly, cotangent and cosecant are contractions of complementi tangens and complementi secans respectively. The abbreviations, sin, cos, tan, cot, sec, and csc did not come into general use until the middle of the 18th century. 2. Express each of the following functions in terms of angles less than 45°: sin 68°, cot 88°, sec 75°, csc 47° 58′ 12", cos 71° 12′ 56′′. In the following right triangles, calculate the required parts from the given parts: Since the functions are equal the angles are equal. = = cot 3 A; find A. 11. Given tan A 12. Given cos ẞ 13. Given tan 20. Given the function of an angle in any quadrant, to construct the angle. - Example 1. Given sin = 3. Construct the angle and find all the other functions. r = 5 units. Draw AB || OX and 3 units above it as in Fig. 25. to . Then from the definitions of the trigonometric functions we Example 2. Given cos 0=-3. Construct the angle and find all the other functions. = 3 X M we take x = 2 units and r units. Draw AB || OY and 2 units to the left as in Fig. 26. Construct a circle of radius 3, with its center at 0, and intersecting AB at P1 and P2. Draw OP1 and OP2. As in Example 1, it may be shown that XOP1 and XOP2 = 02 are the required angles. The remaining functions are as follows: = 01 A FIG. 26. = +3 and x we may take y 4. Then r = √(±4)2+(±3)2=5. With O as a center and 5 as a radius construct a circle as in Fig. 27. Draw AB and CD || OY and 4 units to the right and left respectively of OY. Also draw EF and GH || OX and 3 units above and below OX respectively. These lines and the circle intersect at the points P1, P2, P3, and P4. Since x and y must both be positive or both negative, the required points must be P1 and P3 located in the first and third quadrants. Draw OP, and OP, forming the angles XOP1 = 01 and XOP, = 03. The remaining functions are as follows: In Exercises 1 to 5 tabulate the functions of @ in each of the two quadrants in which the angle is found. 6. What is the greatest value that the sine of an angle may have? That the cosine of an angle may have? What is the least value for each? 7. Between what two numbers will sec @ and csc @ have no values? In Exercises 8 to 10 show by substitution that the right-hand member is equal to the left. 8. cos tan + sin @ cot = sin 0 + cos 0, when tan the third quadrant. 9. (1+tan2 0) (1 the fourth quadrant. == 2 and 0 is in cot2 0) csc2 0, when cos 0 = and is in 1 sin @ = csc 0, when sin 0 = = √3 and 0 is in the second 1 + cos 10. cot @ + quadrant. |