1. Section 3.3 Graphing Sine Cosine and Tangent Functions
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2. I) What is a Periodic Function?
A function that repeats its output values in a cycle
The range of input values within a cycle is known as the period
There are many examples of periodic functions in our world
Ocean waves and ocean currents
Solar system
Finance and Market trends
Sound waves, light transmission, radiation
Wave propagation: earthquakes,
Heart beat
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3. II) Components of a Periodic Function:
½ of Period
Crest
Period x y
Amplitude
Period
Trough
Crest: Local maximums on the periodic function. Top
Trough: Local minimums on the periodic function. Bottom
Amplitude: The vertical difference between the top and the middle
Period: The horizontal difference between two rests or two troughs
Note: The horizontal distance between the crest and trough is HALF the
period.
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4. Ex: Given each graph, indicate the period and amplitudey -3 -2 -1 1 2 -4 3 4 x y -3 -2 -1 1 2 -4 3 4
Amplitude
Amplitude
Period
Period
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5. Ex: If high tide occurred at 7:30am and low tide was at 1:30pm, what time will the next high tide be?
The time difference between high tide and low tide is half the period
The time for a full period will be 12 hours
So the next high tide will be 12 hours from 7:30am
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6. High and Low Tide: At high tide, the ocean levels go up
Ocean levels will fluctuate
up and down over several hours
At low tide, the ocean levels go down

7. III) SINE Function in an Unit Circles:
Sinθ in an unit circle is equal to the y-coordinate of point “P”
The highest y-coordinate is at 1 (at 90°) & the lowest at -1 (at 270°)
The value of Sinθ changes periodically from 1 to -1 depending on the value of the angle
If we graph the sine function, we are comparing how height of point P changes in relation to the angle in standard position
Note: hyp= 1
Y-coordinate
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8. IV) Graphing the Sine Function:x y
-0.5
0.5 1 p 2p 3p 4p -1
X-axis: value of the central angle in radians
Y-axis: value of sinθ, height of point “P”
Highest (1) and lowest (-1)
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9. Sine Function: x y
0.5p p
1.5p 2p -1
-0.5
0.5 1
As you rotate the terminal arm, the value of the angle increases
The height of the graph changes fluctuates up and down in accordance to the angle
The result is a wave function
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10. Ex: Find the following for the sine function: Period & amplitude
X intercept and general formula
Y intercept
Domain and Range
Period x y
-0.5
0.5 p 2p 4p 3p
Amplitude
Period:
Amplitude
X-intercepts:
Y-intercepts:
Domain:
Range:

11. V) COSINE Function in an Unit Circles:
Cosθ in an unit circle is equal to the x-coordinate of point “P”
The lowest x-coordinate is -1 (at 180°) & the highest is 1 (at 360°)
The value of Cosθ changes periodically from 1 to -1 depending on the value of the angle
If we graph the Cosine function, we are comparing the horizontal distance of point P vs the angle in standard position
Note: hyp= 1
x-coordinate
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12. VI) Graphing the Cosine Function:x y
-0.5
0.5 1 p 2p 3p 4p -1
X-axis: value of the central angle in radians
Y-axis: value of cosθ, x-coordinate point “P”
Highest (1) and lowest (-1)
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13. Cosine Function: The result is a wave functionx y
0.5p p
1.5p 2p -1
-0.5
0.5 1
Rotate your unit circle, so you can measure the horizontal distance of point P
The Cosine graph will then begin at the top
Then you graph how the horizontal distance of P changes in relation to the angle in standard position
The result is a wave function
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14. Ex: Find the following for the cosine function: Period & amplitude
X intercept and general formula
Y intercept
Domain and Range
Period x y
-0.5
0.5 p 2p 4p 3p
Amplitude
Period:
Amplitude
X-intercepts:
Y-intercepts:
Domain:
Range:

15. VII) Summary for Graphing Cosine/Sine Functions
When graphing a Sine & Cosine function, there are only 5 points to consider: Beginning, Middle, End, Quarter way, and 3 Quarters way of the period
Sine Function:
At the Beginning, Middle, and End of the period, the wave will cross the x-axis
At quarter way, the graph will be at the top
At 3-quarters way, the graph will be the bottom
Cosine Function
At the Beginning and End of the period, the graph will be at the top
In the Middle, the wave will be at the bottom
At quarter way & 3 quarters way, the graph will be on the x-axis
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16. Ex: What is the equation for the following graph?y
-0.5
0.5 p 2p 4p 3p
The graph is a sine function with a vertical reflection

17. Review: Tangent Function & Tangent Lines
SOH-CAH-TOA
Using the ratio of opposite over adjacent, we can show that the tangent function is also sine over cosine
A tangent line is a line that crosses a circle at only one point
Tangent Line
Point of Tangency
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18. VIII) Tangent Functions and Unit Circles
There are two ways to relate the tangent function with an unit circle
1st Method
The tangent function can be defined as the y-coordinate of Point P, divided by its x-coordinate
2nd Method
The tangent function can be defined as the y-coordinate of the extension of the terminal arm on a vertical tangent line
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19. IX) Tangent = Y-Coordinate / X-Coordinate
The tangent function is the ratio of the Y-coordinate over the X-coordinate
As we rotate the around the unit circle, the ratio changes according to the angle in standard position
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20. X) Tangent as the Y-coordinate on the Vertical Tangent Line
The tangent function can be defined as the y-coordinate of the extension of the terminal arm on a vertical tangent line
When you rotate the terminal arm, the extension creates a right triangle with a base of 1
The tangent function will be the Y-coordinate of this point on the Vertical tangent
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21. As the terminal arm rotates around the circle, it will cross the tangent line above the X-axis (+’ve) or below the X-axis (–’ve )
Ie: when the terminal arm is in Q1, it crosses the tangent line above the X-axis, so tanθ is positive
In Q2, the terminal arm extends to the tangent line beneath the X-axis, so it will be negative
In Q3, it crosses the tangent line above the X- axis, so tanθ is positive
In Q4, it crosses the tangent line under the X- axis, so tan θ is negative
At any point when the terminal arm is pointing side ways, the value of tanθ is equal to zero
At any point when it’s pointing up or down, the value of tanθ is undefined, because it doesn’t touch the tangent line
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22. XI) Tangent function as a Reciprocal
The Tangent function is like a reciprocal function
At points where cosθ = 0, you will get a vertical asymptote because the reciprocal of zero  Undefined
At points where sinθ = 0, you will be a x-intercept
At points where sinθ = cosθ , the value of tanθ will be equal to 1 (ie: 45°, 135°, 225°, and 315°, reference angle equals 45°)
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23. Tangent Function x y p 2p 3p -3 -2 -1 1 2 3 p 2p 3p
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24. The tangent function can be defined as the y-coordinate of the extension of the terminal arm on a vertical tangent line
As the angle changes and increase, the value of tanθ will fluctuate from negative to positive infinity x y
0.5p p
1.5p 2p
2.5p 3p
3.5p 4p
4.5p -2 2
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25. X-intercepts and the general formula Period and amplitude
Ex: Indicate the following for the tangent function:
Domain & Range
X-intercepts and the general formula
Period and amplitude x y p 2p 3p -3 -2 -1 1 2 3 p 2p 3p
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26. XII) Vertical Asymptote and General Formula:
Since tanθ = (sinθ/cosθ), the vertical asymptotes will all appear when cosθ=0 (Denominator is equal to zero)
The general formula for all the vertical asymptotes in a tangent function will be:
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27. Ex: Given that 0< θ< 2π , tanθ > 0 and sinθ < 0, what quadrant is the angle in?
If tanθ > 0 (positive), then it can only be in quadrants 1 and 3
If sinθ < 0 (negative), then it can only be in quadrants 2 and 3
To satisfy both conditions, then the angle must be in quadrant 3
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28. Homework:
Assignment 3.3
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