(1) Trigonometry 5

(1) Trigonometry 5 - Pre – Calculus Math 40S: Explained!...

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Unformatted text preview: Pre – Calculus Math 40S: Explained! www.math40s.com 42 TRIGONOMETRY LESSON FIVE PART I - PERIOD The period of a graph is defined as the length of one complete cycle. This graph has a period of 2π This graph has a period of 4π This graph has a period of π Most graphs given to you won’t be as simple as the first three. In trig graphs that are continuous, you will have to first identify a sine or cosine pattern before you can determine the period. The easiest way to do this is to draw a square around either pattern and look at the length. The graph on the left has a period of π Pre – Calculus Math 40S: Explained! www.math40s.com 43 TRIGONOMETRY LESSON FIVE PART I - PERIOD Questions: For each of the following graphs, draw a rectangle around the indicated pattern and state the period. 1) Draw a rectangle around a sine pattern. 2) Draw a rectangle around a cosine pattern. 3) Draw a rectangle around a sine pattern. 4) Draw a rectangle around a sine pattern. 5) 6) Draw a rectangle around a cosine pattern. Draw a rectangle around a sine pattern. Pre – Calculus Math 40S: Explained! www.math40s.com 44 TRIGONOMETRY LESSON FIVE PART I - PERIOD ANSWERS: 1) Period = 8π 2) Period = 4π 3) Period = 4π 4) Period = 5) Period = π 6) Period = π 3 2π 3 Pre – Calculus Math 40S: Explained! www.math40s.com 45 TRIGONOMETRY LESSON FIVE PART II - THE B VALUE The “b” value represents the number of cycles a trig graph has within a span of 2π. It is the number that you see in a trig function right beside θ. (y = sinbθ) The b value is NOT the period. The b-value and period (for radians) are related by the formula: The b-value and period (for degrees) are related by the formula: Example 1: Draw the graph of y = sin2θ 2π 2π or b = b Period 360 360 Period = or b = b Period Period = (0 ≤ θ ≤ 2π ) The first step in graphing this trig function is to find the period. 2π b 2π Period = 2 Period = π Note that tanθ graphs do not use these equations for the period and b-value. Period = Once we know the period, draw the graph from 0 to 2π, since that is the specified domain. Example 2: Draw the graph of y = cos 1 θ 2 (0 ≤ θ ≤ 4π ) The first step in graphing this trig function is to find the period. 2π b 2π Period = 0.5 Period = 4π Period = Once we know the period, draw the graph from 0 to 4π since that is the specified domain. Pre – Calculus Math 40S: Explained! www.math40s.com 46 TRIGONOMETRY LESSON FIVE PART II - THE B VALUE Given a graph, you must find the b value before you can write the equation. Example 1: Find the cosine equation of the following graph: Step 1: First you need Step 2: Once you identify the Step 3: Now that to draw a rectangle around the cosine pattern. In this graph, we can easily see a cosine patten going period, find b by performing the following calculation: we have the b value, and a cosine pattern is identified, we can write the equation : from 0 to 2π 3 2π Period 2π b= 2π 3 3 b = 2π × 2π b=3 b= y = cos3θ Example 2: Find the sine equation of the following graph: Step 1: First you need to draw a rectangle around the sine pattern you want to use. In this graph, we can easily see a sine pattern going from 0 to 10π Step 2: Once you identify the period, find b by performing the following calculation: 2π Period 2π b= 10 π 1 b= 5 b= Step 3: Now that we have the b value, and we identified a sine pattern, we can write the equation: y = sin 1 θ 5 Pre – Calculus Math 40S: Explained! www.math40s.com 47 TRIGONOMETRY LESSON FIVE PART II - THE B VALUE Questions: For each of the following graphs, write the equation: 1) 2) 3) 4) For each of the following equations, draw the graph: 5) 7) y = sin2θ y = cos3θ 6) 1 y = cos θ 3 8) 1 y = sin θ 4 Pre – Calculus Math 40S: Explained! www.math40s.com 48 TRIGONOMETRY LESSON FIVE PART II - THE B VALUE Answers: 2 5 1) cos4θ b= 2) cos θ 2π 2π 2 = = 2π × = 4 π P π 2 1 2 3) sin θ b= b= 2π 2 π 2 = = P 5π 5 4) sin6θ 2π 2 π 1 = = P 4π 2 b= 2π 2π 3 = = 2π × = 6 π P π 3 2π b 2π P= 1 3 P= 2π b 2π P= 2 P =π P= 5) 6) P = 2π × P = 6π 3 1 2π b 2π P= 1 4 P= 2π b 2π P= 3 P= 7) 8) P = 2π × P = 8π 4 1 Pre – Calculus Math 40S: Explained! www.math40s.com 49 TRIGONOMETRY LESSON FIVE PART III - THE C VALUE The phase shift is the horizontal translation applied to a trig graph. It is the number added or subtracted to θ inside the equation. Phase shift is represented by the letter “c” in y = sin(θ ± c) Notice in the following graphs that you will do the opposite of what the sign is. The + will move the graph left, and the — will move the graph right. y = sinθ y = sin(θ - π ) The -π means we move the graph right by π units. y = sinθ π⎞ ⎛ y = sin ⎜ θ + ⎟ 2⎠ ⎝ π means we 2 move the graph left The + by π 2 units. Not all graphs are going to be given as one cycle, since trig graphs can go forever in both directions! A phase shift will shift everything horizontally by the same amount, so it’s still easy to graph. y = cosθ π⎞ ⎛ y = cos ⎜ θ + ⎟ 2⎠ ⎝ The + π means we 2 move the graph left by π 2 units. Pre – Calculus Math 40S: Explained! www.math40s.com 50 TRIGONOMETRY LESSON FIVE PART III - THE C VALUE Quite often, a graph will be given with ticks where no radian measure is indicated. In these questions, we need to figure out what the exact value of each tick is first. In this graph, we see six ticks between 0 and π, so each tick must zooming in from 0 to π be 30º or In this graph, we see four ticks between 0 and π, Each tick must be zooming in from 0 to π π 45º, or π 6 4 It is always possible to write at least one sine equation and one cosine equation for the same trig graph. Notice how ticks are given in the graph between -π and 0. Think in terms of degrees for a moment. If we have 180º and 6 ticks, that makes each one 30º. So, if our sine pattern starts at the fourth tick back, that would be -120º, or in radians, - 2π ⎛ ⎝ . The sine equation is y = sin ⎜ θ + 2π ⎞ ⎟. ⎠ 3 3 Likewise, we can see that if we were thinking in terms of cosine, the cosine pattern starts one tick back, at -30 º ⎛ ⎝ The cosine equation would be y = cos ⎜ θ + 2π ⎞ ⎛ y = sin ⎜ θ + ⎟ 3⎠ ⎝ ⎞ ⎟ 6⎠ π π⎞ ⎛ y = cos ⎜ θ + ⎟ 6⎠ ⎝ OR Always try to find an “upright” trig pattern to derive the equation. Pre – Calculus Math 40S: Explained! www.math40s.com 51 TRIGONOMETRY LESSON FIVE PART III - THE C VALUE Questions: For 1 & 2, write the sine equation. For 3 & 4, write the cosine equation. 1. 2. 3. 4. For 5 & 6, draw the sine graph. For 7 & 8, draw the cosine graph. π 5. y = sin(θ + ) 6. 3 7. y = cos(θ + π 4 ) 8. y = sin(θ - y = cos(θ - π 4 5π 6 ) ) Pre – Calculus Math 40S: Explained! www.math40s.com 52 TRIGONOMETRY LESSON FIVE PART III - THE C VALUE Answers: 1. sin(θ - π 4 We can draw a rectangle around the sine pattern closest to the origin. There are four ticks between 0 and π, so each one is 45º. Since the sine pattern starts on the first tick to the right, ) the equation is y = sin(θ - 2. 2π sin(θ ) 3 π 4 ) We can draw a rectangle around the sine pattern closest to the origin. There are six ticks between 0 and π, so each one is 30º. Since the sine pattern starts on the fourth tick to the right, which is 120º, the equation is y = sin(θ - 3. cos(θ + 4. cos(θ + π 2 4 3 ) We can draw a rectangle around the cosine pattern closest to the origin. There are six ticks between 0 and π, so each one is 30º. Since the cosine pattern starts on the third tick to the left, ) π 2π which is -90º, the equation is y = cos(θ + ) π 2 ) We can draw a rectangle around the cosine pattern closest to the origin. There are four ticks between 0 and π, so each one is 45º. Since the cosine pattern starts on the first tick to the left, which is -45º, the equation is y = cos(θ + 5. 4 ) 6. 7. π 8. Pre – Calculus Math 40S: Explained! www.math40s.com 53 TRIGONOMETRY LESSON FIVE PART IV - GRAPHING B AND C We will now look at trig graphs with the form: y = sinb(θ + c) ”b” is used to find the period using the formula: Period = 2π b “c” is the letter used to represent phase shift. When combining b & c, we should follow a particular order. First apply the period, then the phase shift. π Example 1: Graph y = sin2(θ - ): 3 y = sinθ y = sin2(θ - y = sin2θ (Remember to use your formula to find the period, which is π) 1 2 Example 2: Graph y = cos (θ + y = cosθ π 4 y = cos π ) 3 (Move the graph two ticks right, since each tick is 30º and we want 60º) ): 1 2 θ (Remember to use the formula to find period, which is 4π) y = cos 1 2 (θ + π 4 ) (Now move the graph one tick left, since each tick is 45º) Sometimes, the b-value is attached to θ inside the brackets. In the equation y = sin(2θ – π 3 ), we MUST factor out the 2 before graphing. The reason for doing this is that we can now easily read off the phase shift. y = sin(2θ y = sin2(θ - π 3 π 6 ) ) When you pull out the 2, divide each term in the original brackets by 2. Pre – Calculus Math 40S: Explained! www.math40s.com 54 TRIGONOMETRY LESSON FIVE PART IV - GRAPHING B AND C Questions: Graph the following equations: 1) y = sin 2 (θ - π ) 3 2 1 3 3) y = cos (x - π ) π 5) y = sin(2θ - ) 3 π 2) y = sin2(θ - ) 4 4) 6) y = cos(2θ - π ) y = cos(4θ + π ) Pre – Calculus Math 40S: Explained! www.math40s.com 55 TRIGONOMETRY LESSON FIVE PART IV - GRAPHING B AND C Now use the formula 1) 3) 5) First graph y = sinθ First graph y = cosθ First graph y = sinθ b to find the period, which is 3π Move the graph three ticks right, since each tick is 30º and we want a shift of 90º The period is 6π Move the graph one tick right to π. The period is π (Don’t forget to factor out the 2) Move the graph right one unit to 30º, or π/6 P= 2π 2) 4) 6) First graph y = sinθ First graph y = cosθ First graph y = cosθ Now use your formula to find the period, which is π. The period is π (Don’t forget to factor out the 2) The period is π/2 (Don’t forget to factor out the 4) Finally move the graph one tick to the right, since each tick is 45º and we want a shift of 45º Move the graph right two units to 90º, or π/2 Move the graph left one unit to -45º, or -π/4 Pre – Calculus Math 40S: Explained! www.math40s.com 56 ...
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