Properties of the sine and cosine functions : Properties of the sine and cosine functions
Slide 3: 1. The Sine and Cosine functions are both periodic with period 2л. 2. The Sine functions is an odd functions since its graph is symmetric with respect to the origin, while the Cosine functions is an even functions since its graph is symmetric with respect to the y axis. 3. The Sine function is: Increasing in the intervals [0, л/2] and [3л/2, 2 л]; and Decreasing in the interval [л/2, 3л/2], over a period of 2л.
Slide 4: 4. The Cosine function is: Increasing in the intervals [л, 2л]; and Decreasing in the intervals [0, л], over a period of 2л. 5. Both the Sine and Cosine functions are continuous functions. 6. The domain of the Sine and Cosine functions is the set of all real numbers, while the range is restricted to the set of real numbers from -1 to 1.
Slide 5: 7. The amplitude of both the sine and cosine functions is 1, since one half of the sum of the lower bound and the upper bound is 1, that is, ½ [ |1| ] + |-1|] = 2/2 0r 1. 8. The maximum and minimum values of the sine and cosine functions are 1 and -1 respectively, which occur alternately midway between the points where the functions is zero. The key values of s in plotting f(s) = sin s and f(s) = cos s over a full period are 0, л/2, л, 3л /2 and 2 л.
Slide 6: Notice that as the terminal point moves counterclockwise on a unit circle, starting at point (1, 0), and the direction of the increasing values of the sine and cosine functions have the following variations .
Slide 7: Remember: If s is a real number, then P(s) is a terminal point that maps the real number s onto a point on the unit circle.
Slide 8: Nothing the Quadrant in which terminal point P(s) lies, you are able to determine the algebraic sign of the Cosine and Sine functions in the different quadrant as indicated in the table below.
Slide 9: This implies that in: Quadrant I - 0 <s < л/2 → 0< sin s<1 0< cos s<1 Quadrant II - л/2 <s < л → 0< sin s<1 -1< cos s<0 Quadrant III - л <s < 3л/2 → -1< sin s<0 -1< cos s<0 Quadrant IV - 3л/2<s < 2л → -1< sin s<0 0< cos s<1