Periodic Functions

1. Periodic Functions Sec. 4.3c

2. Let’s consider… Light is refracted (bent) as it passes through glass. In the figure, is the angle of incidence and is the angle of refraction. The index of refraction is a constant that satisfies the equation If and for a certain piece of flint glass, find the index of refraction.

3. Periodic Functions A function is periodic if there is a positive number c such that for all values of t in the domain of f. The smallest such number c is called the period of the function. What are some common periodic functions??? The sine, cosine, and tangent functions!!! (what are their periods?) Period of sine and cosine: How do we know these??? Period of tangent:

4. General Information about Periodic Functions … Sin, Cos, and Tan Periodic Function = cyclical, repeating function Cycle = one complete pattern Period = horizontal length of one complete pattern Amplitude = (max-min)/2 Phase Shift = horizontal translation – what will this do to our periodic functions? Vertical translation – what would this do to our graphs?

5. Periodic Functions Find each of the following without a calculator. Rewrite: 1. Note: is just a large multiple of …

6. Periodic Functions Find each of the following without a calculator. 2. Rewrite: Note: and wrap to the same point on the unit circle, so the cosine of one is the same as the cosine of the other…

7. Periodic Functions Find each of the following without a calculator. 3. Note: Since the period of the tangent function is rather than , is a large multiple of the period of the tangent function…

8. In groups of two or three, explain to each other why each of the following statements are true. Base your explanations on the unit circle. Remember that –t wraps the same distance as t, but in the opposite direction. t • For any t, the value of cos(t) lies • between –1 and 1, inclusive. t The x-coordinates on the unit circle lie between –1 and 1, and cos(t) is always an x-coordinate on the unit circle.

9. In groups of two or three, explain to each other why each of the following statements are true. Base your explanations on the unit circle. Remember that –t wraps the same distance as t, but in the opposite direction. t 2. For any t, the value of sin(t) lies between –1 and 1, inclusive. t The y-coordinates on the unit circle lie between –1 and 1, and sin(t) is always a y-coordinate on the unit circle.

10. In groups of two or three, explain to each other why each of the following statements are true. Base your explanations on the unit circle. Remember that –t wraps the same distance as t, but in the opposite direction. t 3. The values of cos(t) and cos(–t) are always equal to each other (recall that this is the check for an even function). t The points corresponding to t and –t on the number line are wrapped to points above and below the x-axis with the same x-coordinates  cos(t) and cos(–t) are equal.

11. In groups of two or three, explain to each other why each of the following statements are true. Base your explanations on the unit circle. Remember that –t wraps the same distance as t, but in the opposite direction. t 4. The values of sin(t) and sin(–t) are always opposites of each other (recall that this is the check for an odd function). t The points corresponding to t and –t on the number line are wrapped to points above and below the x-axis with exactly opposite y-coordinates  sin(t) and sin(–t) are opposites.

12. In groups of two or three, explain to each other why each of the following statements are true. Base your explanations on the unit circle. Remember that –t wraps the same distance as t, but in the opposite direction. t 5. The values of sin(t) and sin(t + ) are always equal to each other. t Since is the distance around the unit circle, both t and t + get wrapped to the same point.  This is true for all six trigonometric functions!!!

13. In groups of two or three, explain to each other why each of the following statements are true. Base your explanations on the unit circle. Remember that –t wraps the same distance as t, but in the opposite direction. t 6. The values of sin(t) and sin(t + ) are always opposites of each other (the same is true of cos(t) and cos(t + )). t The points corresponding to t and t + get wrapped to points on either end of a diameter on the unit circle. These points are symmetric with respect to the origin and therefore have coordinates (x, y) and (–x, –y). Therefore sin(t) and sin(t + ) are opposites.

14. In groups of two or three, explain to each other why each of the following statements are true. Base your explanations on the unit circle. Remember that –t wraps the same distance as t, but in the opposite direction. t 7. The values of tan(t) and tan(t + ) are always equal to each other (unless they are both undefined). t By our previous observation, tan(t) and tan(t + ) are ratios of the form and , which are either equal to each other or both undefined.

15. In groups of two or three, explain to each other why each of the following statements are true. Base your explanations on the unit circle. Remember that –t wraps the same distance as t, but in the opposite direction. t 8. The sum always equals 1. t The sum is always of the form for some (x, y) on the unit circle. Since the equation of the unit circle is , the sum is always 1.

16. In groups of two or three, explain to each other why each of the following statements are true. Base your explanations on the unit circle. Remember that –t wraps the same distance as t, but in the opposite direction. t t At this point, we can use reference triangles and quadrantal angles to evaluate trig. functions for all integer multiples of 30 or 45 . This leads us to our 16-point unit circle, which you must commit to memory!!!