Section 2.6 Additional Trigonometric Graphs - Mrsk.ca
Section 2.6 – Additional Trigonometric Graphs Vertical Asymptote A vertical asymptote is a vertical line that the graph approaches but does not intersect, while function values increase or decrease without bound as x-values get closer and closer to the line. Graphing the Tangent Functions The graphs of y A tan Bx C D will have the following characteristics: Domain: x x 2n 1 , where n 2 Range: , The graph is discontinuous at values of x of the form x (2n 1) and has vertical asymptotes at 2 these values. Its x-intercepts are of the form x n . Its period is . Its graph has no amplitude, since there are no minimum or maximum values. The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, tan x tan x . Period No Amplitude B Phase Shift C Vertical translation D B Vertical Asymptote (VA): bx c 2n 1 2 One cycle: 0 argument or argument 2 2 58
Example Find the period, and the phase shift and sketch the graph of y 1 tan x 2 4 Solution Period: P B Phase shift: C 4 B 1 4 Vertical translation: y 0 Vertical Asymptote: x 2n 1 4 x n 4 2 2 x n 4 4 2 4 x n 4 One Complete cycle can be determined by: x x 2 4 2 2 4 4 4 2 4 3 x 4 4 59 x y 1 tan x 0 4 0 1 4 4 0 0.5 4 1 4 2 3 4 4 4 3 4 4 2 2 0.5 0 4
Cotangent Functions Domain: x x n , where n Range: , The graph is discontinuous at values of x of the form x n and has vertical asymptotes at these values. Its x-intercepts are of the form x 2n 1 . 2 Its period is . Its graph has no amplitude, since there are no minimum or maximum values. The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, cot x cot x . Example Find the period, and the phase shift and sketch the graph of y cot 2 x 2 Solution P B 2 Period: Phase shift: C 2 B 2 4 2 2 4 4 y cot 2 x 0 1 0 1 0 4 One cycle: 0 2 x 2 2 x 3 4 4 1 4 2 3 4 4 x 3 V.A: 2 x n 2 x n x n 2 2 2 4 4 60 x 4 4 2 3 4 1 2
Graphing the Secant Function Domain: x x 2n 1 2 , where n Range: , 1 1, The graph is discontinuous at values of x of the form x 2n 1 and has vertical asymptotes at 2 these values. There are no x-intercepts. Its period is 2 . Its graph has no amplitude, since there are no minimum or maximum values. The graph is symmetric with respect to the y-axis, so the function is an even function. For all x in the domain, sec x sec x . y 2cos x x Example Sketch the graph of y 2sec x 4 0 4 2 3 4 0 5 2 3 7 4 4 2 0 2 9 2 4 Solution 2 Period 2 2 1 First, graph y 2cos x 4 4 4 C Phase shift: 4 B 1 4 4 Vertical Asymptote: x 3 , 7 , 11 , 4 2 4 4 4 61 4 4 4
Graphing the Cosecant Function Domain: Range: x x n , where n , 1 1, The graph is discontinuous at values of x of the form x n and has vertical asymptotes at these values. There are no x-intercepts. Its period is 2 . Its graph has no amplitude, since there are no minimum or maximum values. The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain csc x csc x . Example Find the period and sketch the graph of y csc 2 x y csc 2 x y sin 2 x x Solution 0 2 2 1 sin 2 x Period 2 2 4 1 0 2 2 3 0 4 1 0 4 2 First, graph y sin 2 x Phase shift: C B 2 Vertical Asymptote: x 0, , , 2 4 2 2 62 0 2
Finding the Secant and Cosecant Functions from the Graph Example Find an equation y k A sec Bx C or y k A csc Bx C to match the graph 4 2 -1 -0.5 0.5 1 1.5 2 2.5 -2 -4 Solution For cosine: A 2 P 2 2 B 2 B Phase shift C 0 C 0 B 2 y 2 sec( x) from –1 to 2.5. 63
Section 2.6 – Additional Trigonometric Graphs Exercises 1. Find the period, show the asymptotes, and sketch the graph of y tan x 2. Find the period, show the asymptotes, and sketch the graph of y 2 tan 2 x 3. Find the period, show the asymptotes, and sketch the graph of y 1 tan 1 x 4. Find the period, show the asymptotes, and sketch the graph of y cot x 4 2 2 3 5. Find the period, show the asymptotes, and sketch the graph of y 2cot 2 x 6. Find the period, show the asymptotes, and sketch the graph of y 1 cot 1 x 7. Find the period, show the asymptotes, and sketch the graph of y sec x 8. Find the period, show the asymptotes, and sketch the graph of y 2sec 2 x 9. Find the period, show the asymptotes, and sketch the graph of y 3sec 1 x 10. Find the period, show the asymptotes, and sketch the graph of y csc x 11. Find the period, show the asymptotes, and sketch the graph of y 2csc 2 x 12. Find the period, show the asymptotes, and sketch the graph of 13. Graph over a 2-period interval y 1 2cot 2 x 14. Graph over a 2-period interval y 2 tan 3 x 2 15. Graph over a one-period interval y 1 1 csc x 3 16. Graph over a one-period interval y 2 1 sec 1 x 17. 18. 19. 4 4 2 2 2 2 4 2 3 3 2 4 2 2 Graph one complete cycle y 3 2 tan x 2 8 Graph two complete cycles y 2 cot x 4 4 Graph y 1 sec 2 x 3 for 3 x 3 2 2 64 4 2 3 2 y 4csc 1 x 2 4
20. 2 Graph one complete cycle y 1 3csc x 3 4 21. A fire truck parked on the shoulder of a freeway next to a long block wall. The red light on the top is 10 feet from the wall and rotates through one complete revolution every 2 seconds. Graph the function that gives the length d in terms of time t from t 0 to t 2. 22. Find an equation to match the graph 65
23. A rotating beacon is located at point A next to a long wall. The beacon is 9 m from the wall. The distance a is given by a 9 sec 2 t , where t is time measured in seconds since the beacon started rotating. (When t 0, the beacon is aimed at point R.) Find a for t 0.45 24. A rotating beacon is located 3 m south of point R on an east-west wall. d, the length of the light display along the wall from R, is given by d 3tan 2 t , where t is time measured in seconds since the beacon started rotating. (When t 0, the beacon is aimed at point R. When the beacon is aimed to the right of R, the value of d is positive; d is negative if the beacon is aimed to the left of R.) Find a for t 0.8 25. The shortest path for the sun’s rays through Earth’s atmosphere occurs when the sun is directly overhead. Disregarding the curvature of Earth, as the sun moves lower on the horizon, the distance that sunlight passes through the atmosphere increases by a factor of csc , where is the angle of elevation of the sun. This increased distance reduces both the intensity of the sun and the amount of ultraviolet light that reached Earth’s surface. a) Verify that d h csc b) Determine when d 2h c) The atmosphere filters out the ultraviolet light that causes skin to burn, Compare the difference between sunbathing when and when . 3 2 Which measure gives less ultraviolet light? 66
26. Let a person whose eyes are h feet from the ground stand d feet from an object h feet tall, where 1 1 h h feet. Let be the angle of elevation to the top of the object. 2 1 a) Show that d h h cot 2 1 b) Let h 55 and h 5 . Graph d for the interval 0 2 1 2 67
Solutions Section 2.6 – Additional Trigonometric Graphs Exercise Find the period, show the asymptotes, and sketch the graph of y tan x 4 Solution Period 1 Shifted right by 4 x 2 4 2 x 3 4 4 Asymptotes: x n 4 x , , 3 , 7 , 4 4 4 Exercise Find the period, show the asymptotes, and sketch the graph of y 2 tan 2 x Solution Period 2 2x 2 2 2 2 x 0 x 0 2 Asymptotes: x n 2 x , , 0, , , 2 2 83 2
Exercise 2 Find the period, show the asymptotes, and sketch the graph of y 1 tan 1 x 4 3 Solution Period 2 1/ 2 1x 2 2 3 2 x 2 3 5 x 3 3 Asymptotes: x 5 2n 3 x , 5 , , 7 , 3 3 3 Exercise Find the period, show the asymptotes, and sketch the graph of y cot x Solution Period 1 Asymptotes: x 2n 1 4 x 2n 4 x 2n 3 4 x , , 3 , 7 , 4 4 4 84 4
Exercise Find the period, show the asymptotes, and sketch the graph of y 2cot 2 x 2 Solution Period 1 Asymptotes: 2 x 2n 1 2 2 x 2n 2 2 x 2n 2 x n 4 x , , 3 , 7 , 4 4 4 Exercise 2 Find the period, show the asymptotes, and sketch the graph of y 1 cot 1 x 2 Solution Period 2 1/ 2 Asymptotes: 1 x 2n 2 4 1 x 2n 3 2 4 x 4n 3 2 x , , 3 , 11 , 2 2 2 85 4
Exercise Find the period, show the asymptotes, and sketch the graph of y sec x 2 Solution Period 2 x 2 2 2 0 x Asymptotes: x n 2 2 x n x , , 0, , Exercise Find the period, show the asymptotes, and sketch the graph of y 2sec 2 x Solution Period 2 2 Asymptotes: 2 x n 2 2 2x n x n 1 2 x , , 0, , 86 2
Exercise 3 Find the period, show the asymptotes, and sketch the graph of y 3sec 1 x 3 Solution Period 2 6 1/ 3 Asymptotes: 1 x n 3 3 2 x 3n 3 2 x 3n 2 x , 5 , , 7 , 2 2 2 Exercise Find the period, show the asymptotes, and sketch the graph of y csc x Solution Period 2 Asymptotes: x n 2 x n 2 87 2
Exercise 2 Find the period, show the asymptotes, and sketch the graph of y 4csc 1 x 2 4 Find the period, show the asymptotes, and sketch the graph of y 2csc 2 x Solution Period 2 0 2x 2 2x 2 2 x 4 4 Asymptotes: x n 4 2 Exercise Solution Period 2 1/ 2 0 1 x 2 4 1 x 5 4 2 4 x 5 2 2 Asymptotes: x 2 n 2 88
Exercise Graph over a 2-period interval y 1 2cot 2 x 2 Solution P 2 2 1 2 Vertical translation 1 VA n 2 Exercise 4 Graph over a 2-period interval y 2 tan 3 x 2 3 Solution P 3 3 4 4 4 3 4 3 Vertical translation -2 VA 3 n 4 89
Exercise Graph over a one-period interval y 1 1 csc x 3 2 4 Solution P 2 3 4 Vertical translation 1 Exercise 2 Graph over a one-period interval y 2 1 sec 1 x 4 Solution P 2 4 1 2 2 1 2 Vertical translation 1 VA 3 2n 4 90
Exercise 2 8 Graph one complete cycle y 3 2 tan x Solution Period: P 2 1/2 x x y 0 3 1 4 5 3 4 3 5 4 1 2 7 4 3 4 4 Phase shift: 8 1/2 4 4 2 4 4 2 4 Exercise Graph two complete cycles y 2 cot x 4 Solution Period P 1 Phase shift: 4 1 4 VA: x , 5 4 4 Vertical Translation: y 2 4 x y 4 - 2 -3 3 4 -2 3 5 4 91
Exercise Graph y 1 sec 2 x 3 for 3 x 3 2 2 Solution Period 2 2 x y 1 cos 2 x 0 1 3 4 0 1 2 Exercise 2 Graph one complete cycle y 1 3csc x 3 4 Solution x y 3 0 3 -1 3 1 1 -4 3 2 1 2 -1 3 3 3 2 2 3 4 5 2 -1 2 Period 2 4 /2 3 Phase shift: 4 3 2 2 2 2 2 2 2 2 92 3 3 3 4 0 1 3
Exercise A fire truck parked on the shoulder of a freeway next to a long block wall. The red light on the top is 10 feet from the wall and rotates through one complete revolution every 2 seconds. Graph the function that gives the length d in terms of time t from t 0 to t 2. Solution 2 rad / sec t 2 tan d 10 d 10 tan d (t ) 10tan t Period 1 One cycle: 0 t 0 t 1 t d 10 tan t 0 0 1 4 10 1 2 3 4 1 -10 0 Exercise Find an equation to match the graph Solution P 2 0 A 1 1 1 2 y sec x 2 x 2 93
B 2 2 1 P 2 0 C 0 A 1 1 1 2 y sec( x) 2 x 2 B 2 2 1 P 2 0 C 0 A 3 1 2 2 y 2 csc( x) 2 x 2 Exercise A rotating beacon is located at point A next to a long wall. The beacon is 9 m from the wall. The distance a is given by a 9 sec 2 t , where t is time measured in seconds since the beacon started rotating. (When t 0, the beacon is aimed at point R.) Find a for t 0.45 Solution a 9 sec 2 0.45 9 cos 2 0.45 9.5 m Exercise A rotating beacon is located 3 m south of point R on an east-west wall. d, the length of the light display along the wall from R, is given by d 3tan 2 t , where t is time measured in seconds since the beacon started rotating. (When t 0, the beacon is aimed at point R. When the beacon is aimed to the right of R, the value of d is positive; d is negative if the beacon is aimed to the left of R.) Find a for t 0.8 Solution d 3tan 2 (0.8) 9.23 m 94
Exercise The shortest path for the sun’s rays through Earth’s atmosphere occurs when the sun is directly overhead. Disregarding the curvature of Earth, as the sun moves lower on the horizon, the distance that sunlight passes through the atmosphere increases by a factor of csc , where is the angle of elevation of the sun. This increased distance reduces both the intensity of the sun and the amount of ultraviolet light that reached Earth’s surface. a) Verify that d h csc b) Determine when d 2h c) The atmosphere filters out the ultraviolet light that causes skin to burn, Compare the difference between sunbathing when and when . Which measure gives less ultraviolet light? 3 2 Solution a) sin h 1 d csc d h csc (cross-multiplication) b) sin h h 1 d 2h 2 h sin 1 1 2 6 csc 1 2 c) csc 2 3 1.15 3 3 When the distance to the sun is lager , there is less ultraviolet light reaching the earth’s 3 surface. In this case, sunlight passes through 15% more atmosphere. Exercise Let a person whose eyes are h feet from the ground stand d feet from an object h feet tall, where 1 1 h h feet. Let be the angle of elevation to the top of the object. 2 1 a) Show that d h2 h1 cot b) Let h 55 and h 5 . Graph d for the interval 0 2 1 2 Solution h a) h h h 2 1 cot d h d h h cot 2 1 95
b) d 55 5 cot d 50cot 0 2 96
A rotating beacon is located at point A next to a long wall. The beacon is 9 m from the wall. The distance a is given by at 2 , where t is time measured in seconds since the beacon started rotating. (When t R 0, the beacon is aimed at point .) Find a for t 0.45 24. dA rotating beacon is located 3 m south of point R on an east-west wall .
Primary Trigonometric Ratios There are six possible ratios of sides that can be made from the three sides. The three primary trigonometric ratios are sine, cosine and tangent. Primary Trigonometric Ratios Let ABC be a right triangle with \A 6 90 . Then, the three primary trigonometric ratios for \A are: Sine: sinA opposite hypotenuse Cosine .
Oct 18, 2015 · trigonometric functions make it possible to write trigonometric expressions in various equivalent forms, some of which can be significantly easier to work with than others in mathematical applications. Some trigonometric identities are definitions or follow immediately from definitions. Lesson Vocabulary trigonometric identity Lesson
for trigonometric functions can be substituted to allow scientists to analyse data or solve a problem more efficiently. In this chapter, you will explore equivalent trigonometric expressions. Trigonometric Identities Key Terms trigonometric identity Elizabeth Gleadle, of Vancouver, British Columbia, holds the Canadian women’s
defi ned is called a trigonometric identity. In Section 9.1, you used reciprocal identities to fi nd the values of the cosecant, secant, and cotangent functions. These and other fundamental trigonometric identities are listed below. STUDY TIP Note that sin2 θ represents (sin θ)2 and cos2 θ represents (cos θ)2. trigonometric identity, p. 514 .
Section 4.2 Trigonometric Functions: The Unit Circle 295 Evaluating Trigonometric Functions Evaluate the six trigonometric functions at Solution Moving clockwise around the unit circle,it follows that corresponds to the point Now try Exercise 33. Domain and Period of Sine and Cosine The domain of the sine an
Section 6.5 Modeling with Trigonometric Functions 441 Section 6.5 Modeling with Trigonometric Functions Solving right triangles for angles In Section 5.5, we used trigonometry on a right triangle to solve for the sides of a triangle given one side and an additional angle.
www.ncerthelp.com (Visit for all ncert solutions in text and videos, CBSE syllabus, note and many more) Trigonometric Ratios of Some Standard Angles Trigonometric Ratios of Some Special Angles Trigonometric Ratios of Allied Angles Two angles are said to be allied when their sum or difference is either zero or a multiple of 90 .
Use the basic trigonometric identities to verify other identities. Find numerical values of trigonometric functions. 7 ft 5 ft Transform the more complicated side of the equation into the simpler side. Substitute one or more basic trigonometric identities to simplify expressions. Factor or multiply to simplify expressions.
