(1) Trigonometry 9

(1) Trigonometry 9 - Pre – Calculus Math 40S Explained...

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Unformatted text preview: Pre – Calculus Math 40S: Explained! www.math40s.com 82 Trigonometry Lesson 9 Part I - Graphing Other Trig Functions Graphing y = tanθ & y = cotθ y = tanθ The vertical lines you see are called asymptotes. They are places where the graph is undefined. Recall that tanθ = sinθ cosθ and cotθ = cosθ sinθ tanθ is undefined at the angles where cosθ is equal to zero. ⎡ π 3π ⎤ ⎢2, 2 ⎥ ⎣ ⎦ Likewise, cotθ is undefined whenever sinθ is equal to zero. [0, π ] For tanθ, we can see from the graph that It is important you state the general solution of the asymptotes for each graph. the first positive asymptote occurs at a-value: We only use the term amplitude in describing the graphs of sinθ and cosθ. The other four trig graphs are not “closed in”, they go up & down forever. So, we simply call the a-value the vertical stretch. 2 All asymptotes are exactly π units away from each other. π b and b = . The general equation of the asymptotes is: x= π 2 ± nπ y = cotθ b-value & period: IMPORTANT! The period of a basic tanθ or cotθ graph is π, not 2π like sinθ and cosθ. Thus, we have the following formulas: Period = π π Period c-value: No difference from sinθ and cosθ, but remember to move your asymptotes if you shift the graph. d-value: No difference from sinθ and cosθ. We always write the general solution of tanθ & cotθ asymptotes in the following way: For cotθ, we can see from the graph that the first positive asymptote occurs at 0, and all asymptotes are exactly π units away from each other. The general equation of the asymptotes is: x = 0 ± nπ , or simply, x = ± nπ x = Angle of first positive asymptote ± n(Period) Pre – Calculus Math 40S: Explained! www.math40s.com 83 Trigonometry Lesson 9 Part I - Graphing Other Trig Functions Graphing y = cscθ & y = secθ y = cscθ Grey = sinθ As with the previous graphs, the vertical lines you see are asymptotes. and they occur where the graph is undefined. Recall that csc θ = 1 sin θ and secθ = 1 cosθ At the angles where cosθ is equal to zero, the graph is undefined Likewise, cotθ is undefined whenever sinθ is equal to zero. For cscθ, we can see from the graph that the first positive asymptote occurs at 0, and all asymptotes are exactly π units away from each other. The general equation of the asymptotes is: a-value: Vertical stretch, makes graphs taller and more narrow. b-value & period: For the reciprocal graphs cscθ and secθ, they have a period of 2π, so we can use the formulas we’re used to: Period = 2π b and b = x = 0 ± nπ , or simply, x = nπ 2π Period y = secθ c-value: No difference from sinθ and cosθ, but remember to move your asymptotes if you shift the graph. Grey = cosθ d-value: No difference from sinθ and cosθ. For secθ, we can see from the graph that the first positive asymptote occurs at π 2 All asymptotes are exactly π units away from each other. . The general equation of the asymptotes is: x= We always write the general solution of cscθ & secθ asymptotes in the following way: π 2 ± nπ ⎛ Period ⎞ ⎟ ⎝2⎠ x = Angle of first positive asymptote ± n ⎜ Pre – Calculus Math 40S: Explained! www.math40s.com 84 Trigonometry Lesson 9 Part I - Graphing Other Trig Functions Find the equation of each of the following graphs. Also, state the general solution of the asymptotes. 1. 2. 3. 4. Draw the graph for each of the following equations, and state the general solution of the asymptotes. 5. y = cot3θ 1 7. y = tan θ 2 1 6. y = sec θ 3 8. y = csc2θ Pre – Calculus Math 40S: Explained! www.math40s.com 85 Trigonometry Lesson 9 Part I - Graphing Other Trig Functions 1) The period of this secθ graph is π. (The length of one complete cycle must contain both the top U and the upside-down U.) 2π Now find the b-value: b = = Period General Solution: x = π ±n 4 2π π Equation is: y = sec2θ. = 2. π 2 2) The period of this tanθ graph is π/3. We can tell since each tick is 30º and one cycle uses two ticks. π Now find the b-value: b = = Period π π =π× 3 π = 3. Equation is y = tan3θ . 3 General Solution: x = π 6 ⎛π ⎞ ⎟ ⎝3⎠ ± n⎜ π 3) The period of the cotθ graph is 2π. Now find the b-value: b = General Solution: Period x = ± n ( 2π ) 2π 4) The period of the cscθ graph is 6π. Now find the b-value: b = General Solution: Period x = ± n ( 3π ) 5) = = π 2π 2π 6π 1 = 1 = 1 y = csc θ 3 3 2 1 1 −π π −6 π−5 π−4 π−3 π−2 π −π 1 π 2 π 3 π 4π 5 π 6π 1 General Solution: ⎛π ⎞ x =±n⎜ ⎟ ⎝ 3⎠ 2 3 2 3 8) 3 General Solution: 3π ± n( 3π ) x= 2 3 2 2 1 −4 π −3 π −2 π Equation is 3 2 7) 1 y = cot θ 2 2 6) 3 Equation is 1 −π π 2π 3π 4π −π 1 2 3 π 1 General Solution: x =π ± n( 2π ) 2 3 General Solution: nπ x =± 2 Pre – Calculus Math 40S: Explained! www.math40s.com 86 Trigonometry Lesson 9 Part II - Summary of Trig Functions 1 Example 1: Given the equation: y = csc θ , find the following: 2 Questions: a) a-value b) b-value Answers: a) 1 b) ½ c) Period c) Period = d) Phase shift e) Vertical Displacement f) Domain g) Range h) x-intercepts i) y-intercepts j) Equation of asymptotes d) None e) None f) x ≠ ± n(2π ) g) y ≤ − 1 , y ≥ 1 h) None i) None j) x = ± n(2π ) 3 2π 2π = = 4π b 0.5 Example 2: Given the equation: h(t) = 7sin Questions: a) a-value b) b-value c) Period d) Phase shift e) Vertical Displacement f) Domain g) Range h) x-intercepts i) y-intercepts j) Equation of asymptotes Answers: a) 7 2π b) 5 c) 5 d) 1.25 right e)15 up f) t ∈ R g) 8 ≤ h(t ) ≤ 22 h) None i) 8 j) None 2 1 −4 π −3π −2 π −π π 2π 3π 4π 1 2 3 2π (t -1.25)+15 , find the following: 5 2 5 2 0 1 5 1 0 5 2 4 6 8 1 0 Reminder: To find x-intercepts, use 2nd Trace Zero. You should always state general solutions for x-intercepts. (unless the domain is specified) Find y-intercepts using: 2nd Trace Value x=0 Pre – Calculus Math 40S: Explained! www.math40s.com 87 1 www.math40s.com 10 9. 8. 7. 6. 5. 4. 3. 2. 1. Equation a-value b-value Period Phase Shift Vertical Displacement Domain Range x-intercepts y-intercepts Equation Of Asymptote Trigonometry Lesson 9 Part II - Summary of Trig Functions Pre – Calculus Math 40S: Explained! 88 Trigonometry Lesson 9 Part II - Summary of Trig Functions avalue 1. b-value 2 4 Period Phase Shift π π 2 4 right Vertical Displacement Domain Range x-int y-int Equation Of Asymptotes 5 up θ ∈R 3≤ y ≤7 None 5 None d ∈R −15.5 ≤ T ≤ 26.5 21 2π 365 365 118 right 5.5 up 3. 1 1 2 2π None None 4. 1 2 2 π 45º right 1 up θ ∈R 5. 18.5 2π 365 365 28 right 4.5 up t∈R 6. 1 2 1 3 6π 3 up θ ∈R 7. 1 1 3 6π None None 2. π 2 left x ≠ π ± n ( 2π ) x≠ 3π ± n (3π ) 2 x ≠ ± n(2π ) 316 ± n(360 ) -13.31 None x = π ± n ( 2π ) y∈R ± n(2π ) 0 0 .5 ≤ y ≤ 1.5 None 1 None 303 ± n(360 ) 20.9 None None 3.25 None None 1 y∈R π ± n(2π ) None x = ± n ( 2π ) None 0.29 None 19.6 None −14 ≤ h ≤ 23 2.5 ≤ y ≤ 3.5 134 ± n(360 ) y ≤ −1 & y ≥1 8. 1 1 2 2π None None 9. 2 1 3 6π π right 2 up θ ∈R 0≤ y≤4 20.1 2π 300 300 265 right 6.2 up t∈R −13.9 ≤ y ≤ 26.3 10 102 ± n(360 ) x= 3π ± n (3π ) 2 130 ± n(360 ) 250 ± n(360 ) Pre – Calculus Math 40S: Explained! www.math40s.com 89 ...
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This note was uploaded on 02/09/2011 for the course 168 comm 168 taught by Professor Kiscabean during the Winter '10 term at UCLA.

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