Trigonometric Functions Section 1.6
Quick Review
Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle.
Radian Measure An angle of measure θ is placed in standard position at the center of circle of radius r,
Trigonometric Functions of θ The six basic trigonometric functions of are defined as follows: sine: sin y r cosecant: csc r y cosine: cos x r secant: sec r x tangent: tan y x cotangent: cot x y
Graphs of Six Trigonometric Functions
AP Angle Convention Angle Convention: Use Radians From now on in this book, it is assumed that all angles are measured in radians unless degrees or some other unit is stated explicitly. When we talk about the angle we 3 mean radians (which is 60 ), not degrees. 3 3 When you do calculus, keep your calculator in radian mode.
Periodic Function, Period A function f x is periodic if there is a positive number p such that f x p f x for every value of x. The smallest value of p is the period of f. The functions cos x, sin x, sec x and csc x are periodic with period 2. The functions tan x and cot x are periodic with period.
Even and Odd Trig Functions A function f x is periodic if there is a positive number p such that f x p f x for every value of x. The smallest value of p is the period of f. The functions cos x, sin x, sec x and csc x are periodic with period 2. The functions tan x and cot x are periodic with period.
Transformations of Trig Graphs The rules for shifting, stretching, shrinking and reflecting the graph of a function apply to the trigonometric functions. Vertical Stretch/Shrink Reflection about x-axis Vertical shift y a f b x cd Horizontal stretch or shrink Reflection about the y-axis Horizontal shift
Example: Transformations of Trigonometric Graphs Determine the period, domain, range and draw the graph of y 2sin 4x We can rewrite the function as y 2sin 4x 4 2 The period of y asin bx is. In our example b 4, b 2 so the period is =. The domain is (, ). 4 2 The graph is a basic sin x curve with an amplitude of 2. Thus, the range is [ 2, 2]. The graph of the function is shown together with the graph of the sin x function.
Inverse Trigonometric Functions None of the six basic trigonometric functions graphed in Figure 1.42 is one-toone. These functions do not have inverses. However, in each case, the domain can be restricted to produce a new function that does have an inverse. The domains and ranges of the inverse trigonometric functions become part of their definitions.
Inverse Trigonometric Functions Function Domain Range 1 y cos x 1 x1 0 y 1 y sin x 1 x1 y 2 2 1 y tan x x y 2 2 1 y sec x x 1 0 y, y 2 1 y csc x x 1 y, y 0 2 2 1 y cot x x 0 y
Inverse Trigonometric Functions The graphs of the six inverse trigonometric functions are shown here.
Example Inverse Trigonometric Functions 1 1 Find the measure of sin in degrees and in radians. 2 1 2 The calculator returns 30. 1 1 Put the calculator in radian mode and enter sin. 2 The calculator returns.52359877556 radians. 1 Put the calculator in degree mode and enter sin. This is the same as radians. 6
Quick Quiz Sections 1.4 1.6 You should solve the following problems without using a graphing caluclator. 1. Which of the following is the domain of f ( x) log x 3? (A), (B),3 (C) 3, (D) [ 3, ) (E) (,3] 2 Slide 1.6-17
Quick Quiz Sections 1.4 1.6 You should solve the following problems without using a graphing caluclator. 1. Which of the following is the domain of f ( x) log x 3? ( A), (B),3 (C) 3, (D) [ 3, ) (E) (,3] 2 Slide 1.6-18
Quick Quiz Sections 1.4 1.6 2. Which of the following is the range of f ( x) 5cos x 3? (A), (B) 2,4 (C) 8, 2 (D) 2,8 2 8 (E), 5 5 Slide 1.6-19
Quick Quiz Sections 1.4 1.6 2. Which of the following is the range of f ( x) 5cos x 3? (A), (B) 2,4 (C) 8, 2 (D) 2,8 2 8 (E), 5 5 Slide 1.6-20
Quick Quiz Sections 1.4 1.6 3 3. Which of the following gives the solution of tan x 1 in x? 2 (A) 4 (B) 4 (C) 3 (D) (E) 3 4 5 4 Slide 1.6-21
Quick Quiz Sections 1.4 1.6 3 3. Which of the following gives the solution of tan x 1 in x? 2 (A) 4 (B) 4 (C) 3 (D) (E) 3 4 5 4 Slide 1.6-22
Chapter Test In Exercises 1 and 2, write an equation for the specified line. 1. through 4, 12 and parallel to 4x 3y12 2. the line y f x where f has the following values: x 2 2 4 f(x) 4 2 1 Slide 1.6-23 3. Determine whether the graph of the function y x symmetric about the y-axis, the origin or neither. 4 x 1 4. Determine whether the function y is even, odd or neither. 3 x 2x 1 5 is
Chapter Test In Exercises 5 and 6, find the a domain and b range, and c graph the function. 5. y 2sin 3x 1 6. yln x3 1 Slide 1.6-24
Chapter Test 7. Write a piecewise formula for the function. 1 1 2 Slide 1.6-25
Chapter Test 8. x 5cos t, y 2sin t, 0 t 2 is a parametrization for a curve. a Graph the curve. Identify the initial and terminal points, if any. Indicate the direction in which the curve is traced. b Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? Slide 1.6-26
Chapter Test 9. Give one parametrization for the line segment with endpoints 2, 5 and 4,3. 10. Given f x 2 3 x, 1 1 1 a find f and show that f f x f f x 1 b graph f and f in the same viewing window Slide 1.6-27
Chapter Test Solutions In Exercises 1 and 2, write an equation for the specified line. 4 20 1. through 4, 12 and parallel to 4x 3y 12 y x 3 3 2. the line y f x where f has the following values: x 2 2 4 f(x) 4 2 1 1 y x 3 2 3. Determine whether the graph of the function y x symmetric about the y-axis, the origin or neither. 1 5 is Origin 4 x 1 4. Determine whether the function y is even, odd or neither. Od d 3 x 2x Slide 1.6-28
Chapter Test Solutions cgraph the function. 5. y 2 x y x a All R In Exercises 5 and 6, find the a domain and b range, and sin 3 1 6. ln 3 1 eals b [ 3, 1] a 3, b All Reals [π, π] by [-5, 5] [2, 10] by [2, 5] Slide 1.6-29
Chapter Test Solutions 7. Write a piecewise formula for the function. 1 f x 1 2 1 x, 0 x1 2 x, 1 x 2 Slide 1.6-30
Chapter Test Solutions 8. x 5cos t, y 2sin t, 0 t 2 is a parametrization for a curve. a Graph the curve. Identify the initial and terminal points, if any. Indicate the direction in which the curve is traced. Initial Point (5, 0) Terminal Point (5, 0) Slide 1.6-31
Chapter Test Solutions 8. b Find a Cartesian equation for a curve that contains the parametrized curve. 2 2 x y 5 2 1 What portion of the graph of the Cartesian equation is traced by the parametrized curve? All Slide 1.6-32
Chapter Test Solutions 9. Give one parametrization for the line segment with x endpoints 2, 5 and 4,3. x 2 6t, y 5 2 t, 0t1 10. Given f 2 3 x, 1 1 1 a find f and show that f f x f f x a 2 3 (A possible answer) 1 2 x f x 3 1 2x 2x f f x f 2 3 2 2 x x 3 3 1 1 f f x f x x 2 2 3x 3x 3 3 Slide 1.6-33
Chapter Test Solutions 1 f f 10. b graph and in the same viewing window f x 2 1 3 x [5, 5] by [5, 5] 23 f x x Slide 1.6-34