Likewise, immediately before a becomes 90°, tan ɑ or y х is a very large positive number; while immediately after a has passed 90°, tan a is a very large negative number. It has been seen that tan 90°∞. Hence tan a passes through and changes sign as a passes through 90°. Hence we may say that tan 90° = ±∞, choosing the positive sign when associating 90° with the first quadrant and the negative sign when associating 90° with the second quadrant. Similarly, whenever a trigonometric function passes through ∞ it changes sign. b. As the angle a increases from 90° to 180°, CSC α or y is positive and increases from 1 to + ∞ c. As the angle a increases through 180°, y passes through zero, changing from a positive number to a negative number. As the angle a increases from 180° to 270°, d. As the angle a increases through 270°, x passes through zero, changing from a negative number to a positive number. As the angle a increases from 270° to 360°, 1st QUADRANT 2d QUADRANT | 3d QUADRANT | 4th QUADRANT 180° 270° - 0 dec. 1 - 1 1 inc. х It is thus seen that the sine and cosine can never be greater than +1 nor less than 1, while the secant and cosecant have no values between +1 and -1, but have values ranging from +1 to +∞ and from The tangent and cotangent may have value from +∞ to any 41. Graphical representation. A graphical representation of the trigonometric functions is effected by first locating points using the different values of the angle as abscissas and the corresponding function-values as ordinates, and then drawing a smooth curve through these points taken in the order of increasing angles. The values of the functions of the angles previously calculated are sufficient to determine an approximate graph. For greater accuracy the values of the functions may be taken from the table of natural functions. - 1 to 81. α sin a The sine and tangent curves illustrate the truth of the theorems of Arts. 25 and 26. 3 m Thus for any given value of the angle, as π there is but 4' one value of each function, the curves showing the sine and tangent to be√2 and − 1 respectively. But for any given value of a trigonometric function, as sin α = 1, there are an unlimited number of angles, as Also for tan α = √3 we see that a may have the values π 3' 1, 2, 3π, etc. 42. Periodicity of the trigonometric functions. From the angle is 0, and that it increases to 1 as the angle increases to π 2 At this point the sine begins to decrease and has the value 0 when the angle is 7, and finally reaches the value -1 when the angle is 3; then the sine again begins to in 2 π π crease and has the value 0 when the angle has the value 2. When the angle increases from 2 to 4 π, the sine repeats its values of the interval from 0 to 2 π. If the angle were increased indefinitely, the sine would repeat its values for each interval of 2π. For this reason the sine is called a periodic function, and 2 is its period. A study of the tangent curve shows that the tangent has the same values between 0 and that it has between and 2π or between 2 and 3π. Hence the tangent is also a periodic function having the period. 1. Plot y= cos x and give the period of the cosine. 2. Plot y=cot x and give the period of the cotangent. 3. Plot y = sec x and give the period of the secant. 4. Plot y = cscx and give the period of the cosecant. = sin x cos x. 5. Plot y: 6. Plot y = x + sin x. |