Section 3.3 Graphing Sine Cosine and Tangent Functions © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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. © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Given each graph, indicate the period and amplitude y -3 -2 -1 1 2 -4 3 4 x y -3 -2 -1 1 2 -4 3 4 Amplitude Amplitude Period Period © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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) © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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:

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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) © Copyright all rights reserved to Homework depot: www.BCMath.ca

Cosine Function: The result is a wave function x 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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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:

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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°) © Copyright all rights reserved to Homework depot: www.BCMath.ca

Tangent Function x y p 2p 3p -3 -2 -1 1 2 3 p 2p 3p © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

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: © Copyright all rights reserved to Homework depot: www.BCMath.ca

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 © Copyright all rights reserved to Homework depot: www.BCMath.ca

Homework: Assignment 3.3 © Copyright all rights reserved to Homework depot: www.BCMath.ca