In Exercises 1-12, graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function.

Size: px

Start display at page:

Download "In Exercises 1-12, graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the function."

Theresa Gray
5 years ago
Views:

1 0.5 Graphs of the Trigonometric Functions Eercises In Eercises -, graph one ccle of the given function. State the period, amplitude, phase shift and vertical shift of the function.. = sin. = sin. = cos. = cos 5. = sin +. = sin 7. = cos + 8. = cos + 9. = sin 0. = cos +. = cos +. = sin + In Eercises -, graph one ccle of the given function. State the period of the function.. = tan. = tan 5. = tan +. = sec 7. = csc + 8. = sec + 9. = csc 0. = sec +. = csc. = cot +. = cot 5. = cot + + In Eercises 5 -, use Eample 0.5. as a guide to show that the function is a sinusoid b rewriting it in the forms C = A cosω + φ + B and S = A sinω + φ + B for ω > 0 and 0 φ <. 5. f = sin + cos +. f = sin cos 7. f = sin + cos 8. f = sin cos 9. f = cos sin 0. f = cos sin +. f = cos5 sin5. f = cos sin

2 80 Foundations of Trigonometr. f = 5 sin 5 cos. f = sin cos 5. In Eercises 5 -, ou should have noticed a relationship between the phases φ for the S and C. Show that if f = A sinω + α + B, then f = A cosω + β + B where β = α.. Let φ be an angle measured in radians and let P a, b be a point on the terminal side of φ when it is drawn in standard position. Use Theorem 0. and the sum identit for sine in Theorem 0.5 to show that f = a sinω + b cosω + B with ω > 0 can be rewritten as f = a + b sinω + φ + B. 7. With the help of our classmates, epress the domains of the functions in Eamples 0.5. and using etended interval notation. We will revisit this in Section 0.7. In Eercises 8 -, verif the identit b graphing the right and left hand sides on a calculator. 8. sin + cos = 9. sec tan = 0. cos = sin. tan + = tan. sin = sin cos. tan = sin + cos In Eercises - 50, graph the function with the help of our calculator and discuss the given questions with our classmates.. f = cos + sin. Is this function periodic? If so, what is the period? 5. f = sin. What appears to be the horizontal asmptote of the graph?. f = sin. Graph = ± on the same set of aes and describe the behavior of f. 7. f = sin. What s happening as 0? 8. f = tan. Graph = on the same set of aes and describe the behavior of f. 9. f = e 0. cos + sin. Graph = ±e 0. on the same set of aes and describe the behavior of f. 50. f = e 0. cos + sin. Graph = ±e 0. on the same set of aes and describe the behavior of f. 5. Show that a constant function f is periodic b showing that f + 7 = f for all real numbers. Then show that f has no period b showing that ou cannot find a smallest number p such that f + p = f for all real numbers. Said another wa, show that f + p = f for all real numbers for ALL values of p > 0, so no smallest value eists to satisf the definition of period.

3 0.5 Graphs of the Trigonometric Functions Answers. = sin Period: Amplitude: Phase Shift: 0. = sin Period: Phase Shift: 0. = cos Period: Amplitude: Phase Shift: 0. = cos Period: Phase Shift: 5

4 8 Foundations of Trigonometr 5. = sin + Period: Phase Shift: 7 5. = sin Period: Phase Shift: 5 7. = cos + Period: Phase Shift: = cos + Period: Phase Shift: Vertical Shift: 5 5 7

5 0.5 Graphs of the Trigonometric Functions 8 9. = sin Period: Phase Shift: You need to use = sin + to find this Vertical Shift: 0. = cos + Period: 5 Amplitude: Phase Shift: You need to use 8 = cos + to find this. Vertical Shift: = cos + Period: Amplitude: Phase Shift: 7 5 Vertical Shift:. = sin + Period: Amplitude: Phase Shift: You need to use = sin to find this Two ccles of the graph are shown to illustrate the discrepanc discussed on page 79. Again, we graph two ccles to illustrate the discrepanc discussed on page This will be the last time we graph two ccles to illustrate the discrepanc discussed on page 79.

6 8 Foundations of Trigonometr. = tan Period: 7 5. = tan Period: 5 5. = tan + is equivalent to = tan + + via the Even / Odd identit for tangent. Period: 5 8 8

7 0.5 Graphs of the Trigonometric Functions 85. = sec Start with = cos Period: 5 7. = csc + Start with = sin + Period: = sec + Start with = cos + Period: 7 0

8 8 Foundations of Trigonometr 9. = csc Start with = sin Period: 5 0. = sec + Start with = cos + Period: = csc Start with = sin Period: 5 7

9 0.5 Graphs of the Trigonometric Functions 87. = cot + Period: 7 5. = cot 5 Period: = cot + + Period: 5 8 8

10 88 Foundations of Trigonometr 5. f = sin + cos + = sin + + = cos + 7. f = sin cos = sin + 7. f = sin + cos = sin + 8. f = sin cos = sin + 9. f = cos sin = sin 0. f = cos. f = cos5 + = cos + + = cos + sin + = sin + 5 sin5 = sin f = cos sin = sin. f = 5 sin 5 cos = 5 sin + 7. f = sin cos = sin + 5 = cos + 5 = cos = cos + + = cos 5 + = cos + 5 = 5 cos + 5 = cos + 7

Section 7.1 Graphs of Sine and Cosine

Section 7.1 Graphs of Sine and Cosine OBJECTIVE 1: Understanding the Graph of the Sine Function and its Properties In Chapter 7, we will use a rectangular coordinate system for a different purpose. We