(1) Trigonometry 6

(1) Trigonometry 6 - Pre – Calculus Math 40S: Explained!...

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Unformatted text preview: Pre – Calculus Math 40S: Explained! www.math40s.com 57 Trigonometry Lesson 6 Part I ‐ Graphing ABCD Now that we have looked at all the transformations separately, we can combine them together. y = asinb(θ ± c) ± d When drawing a graph from an equation, we must pay close attention to the order. First we apply the stretches (a & b, in either order), then the translations (c & d, in either order). 1 π Example 1: Graph y = 2cos (θ - )+ 3 2 4 First graph y = cosθ Now graph y = cos 3 3 2 2 1 1 −4 π −3 π −2 π −π π 2 π 3 π 4π 5 π −4 π −3 π −2 π −π π 1 1 θ 2 2 3 2 π 3 π 4π 5 π 1 2 Next graph y = 2cos 1 θ 2 3 Now do the phase shift and move the graph one unit right. 1 π y = 2cos (θ - ) 2 4 3 2 1 1 π y = 2cos (θ - )+ 3 2 4 3 2 Finally, do the vertical translation 6 1 4 2 −4 π −3 π −2 π −π π 2 π 3 π 4π 5 π −4 π −3 π −2 π −π π 1 1 2 2 3 2 π 3 π 4π 5 π 3 −4 π −3 π −2π −π π 2π 3 π 4π 5 π 2 Pre – Calculus Math 40S: Explained! www.math40s.com 58 Trigonometry Lesson 6 Part I ‐ Graphing ABCD Questions: Graph each of the following equations. You may find it useful to draw each transformation in a different color. Remember to factor out numbers attached to θ . π 1) y = 2cos3(θ - )- 2 2 2 1 −2 π −π π 2π -1 -2 -3 -4 -5 -6 2) y = -2sin(4θ - π )+ 5 6 4 2 −2 π −π π 2π -2 Pre – Calculus Math 40S: Explained! www.math40s.com 59 Trigonometry Lesson 6 Part I ‐ Graphing ABCD Questions: π 3) y = -5cos(2θ - )+1 3 6 4 2 −2 π −π π 2π -2 -4 θπ 4) y = 2sin( - )+ 2 33 -6 4 3 2 1 −2 π −π π 2π 3π 4π 5π 6π 7π -1 -2 Pre – Calculus Math 40S: Explained! www.math40s.com 60 Trigonometry Lesson 6 Part I ‐ Graphing ABCD Questions: π 5) y = 3cos2(θ - )+ 2 4 4 2 −2 π −π π 2π -2 6) y = - 1 cos(2θ - 90o )+1 2 Pre – Calculus Math 40S: Explained! www.math40s.com 61 Trigonometry Lesson 6 Part I ‐ Graphing ABCD Then graph y = cos3θ (Period = 2π/3) . Answers 1. First graph y = cosθ y = 2cos3(θ - Now graph y = 2cos3θ π ) 2 Move the graph two ticks right. y = 2cos3(θ - π )-2 2 2. First graph y = sinθ y = -2sin4θ Reflect in x-axis y = sin4θ (Period = π/2) y = -2sin4(θ - π y = 2sin4θ ) y = -2sin4(θ - 4 π )+5 4 Pre – Calculus Math 40S: Explained! www.math40s.com 62 Trigonometry Lesson 6 Part I ‐ Graphing ABCD Answers: 3. y = cos2θ y = cosθ y = -5cos2θ y = -5cos2(θ - y = 5cos2θ π y = -5cos2(θ - ) 6 4. y = sinθ y = sin 1 1 θ (θ - π ) y = 2sin 1 θ 3 y = 2sin 3 ) +1 6 3 y = 2sin π 1 (θ - π ) + 2 3 Pre – Calculus Math 40S: Explained! www.math40s.com 63 Trigonometry Lesson 6 Part I ‐ Graphing ABCD y = cos2θ Answers 5. y = 3cos2θ y = cosθ y = 3cos2(θ - π 3cos2(θ - ) 4 6. y = cosθ π )+ 2 4 y = cos2θ y= 1 cos2θ 2 y=- 1 2 cos2θ y=- 1 o cos2(θ - 45 ) 2 y=- 1 o cos2(θ - 45 ) + 1 2 Pre – Calculus Math 40S: Explained! www.math40s.com 64 Trigonometry Lesson 6 Part 2 ‐ Inverse Trig Functions Inverse Trigonometric Functions: Memorize the following graphs and their characteristic properties. Pre – Calculus Math 40S: Explained! www.math40s.com 65 Trigonometry Lesson 6 Part 2 ‐ Inverse Trig Functions ⎛ 3⎞ Example 1: Evaluate sin -1 ⎜ ⎜2⎟ ⎟ ⎝ ⎠ 3 2π 3 , and sin . = 3 2 3 2 π 2π Therefore, we would assume the answer is , . 33 2π However, since is outside the range of the graph y = sin −1 x , we exclude it. 3 π The only solution is . 3 ⎛ 3⎞ *Alternatively, you can type sin −1 ⎜ ⎜ 2 ⎟ in your calculator (in degree mode) to get 60°. ⎟ ⎝ ⎠ We know from the unit circle that sin π = 5π ⎞ ⎛ Example 2: Evaluate cos-1 ⎜ cos ⎟ 4⎠ ⎝ ⎛ 5π ⎞ 2⎞ ⎛ -1 cos -1 ⎜ cos ⎟ = cos ⎜ − ⎜ 2⎟ ⎟ 4⎠ ⎝ ⎝ ⎠ 3π 2 5π 2 From the unit circle, we know that cos =− and cos =− . 4 2 4 2 5π 3π is outside the range of the graph y = cos −1 x , so the only answer is . However, 4 4 ⎛ 2⎞ *Alternatively, you can type cos −1 ⎜ − ⎜ 2 ⎟ in your calculator (in degree mode) to get 135°. ⎟ ⎝ ⎠ Questions: Evaluate the following: (Hint: Remember to use solutions that exist on the inverse graph). ⎛ 3⎞ 1) sin −1 ⎜ − ⎜ 2⎟ ⎟ ⎝ ⎠ 3π ⎞ ⎛ 5) sin −1 ⎜ cos ⎟ 2⎠ ⎝ 2) cos −1 ( 0 ) 3) sin −1 Answers: ⎛ π⎞ 6) cos ⎜ sin ⎟ 4⎠ ⎝ ( −1) −1 ⎛ 1⎞ 4) cos −1 ⎜ − ⎟ ⎝ 2⎠ 1) − 2) π 2 3) − π 2π 3 5) 0 3 4) π 6) 2 π 4 Pre – Calculus Math 40S: Explained! www.math40s.com 66 ...
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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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