Secondary Math GRAPHING TANGENT AND RECIPROCAL TRIG FUNCTIONS/SYMMETRY AND PERIODICITY

Transcription

1 Secondary Math GRAPHING TANGENT AND RECIPROCAL TRIG FUNCTIONS/SYMMETRY AND PERIODICITY

2 Warm Up Factor completely, include the imaginary numbers if any. (Go to your notes for Unit 2)

3 Graphing Tangent To graph f(x) = tanx, we look at the relationship between two similar triangles drawn at the left. The smaller triangle is inscribed in the unit circle and intersects the circle at some point P(x, y). The larger triangle shares a vertex with the smaller triangle, but its shorter leg is the length of the radius of the circle. Setting up a proportion we get = =, thus =. Remember that y = sin and x =!" and #$% = &' (' =. Thus, tangent of is equal to the length of the longer leg of the larger triangle, w.

4 Graphing Tangent If we draw similar triangles for different angles on the unit circle, then we would be able to plot the graph of the tangent function where x = and y = #$%. Recall that tangent is undefined when x = ) and x = ), so the graph will have vertical asymptotes at those values.

5 General Equation and Graph The general equation for tangent is: * = atan. +/ The domain is all real numbers except odd multiples of ). The range is the set of all real numbers. The graph crosses the y-axis half way between the two asymptotes.

6 Vocabulary a represents the vertical stretch or shrink The period, ), is the interval length needed to 0 complete one cycle. * = atan. +/ Frequency, 0, is the number of complete ) cycles a periodic function makes in a specific interval. The asymptotes are determined by: The first set, centered around the origin is given by = 0± ) 0. To determine the remaining asymptotes, add the period to the previous asymptote.

7 Example Identify the period, vertical asymptotes, y-intercept, then sketch one period of the graph. * = #$% Find the period Period is ) 0 = ) 2 3 = 24 Find the asymptotes = 0± ) 0 = 0± ) = ±4 2 3 The x-coordinate of the y-intercept is located halfway between ±4 and the y coordinate is the vertical shift. (0, 0)

8 Example Identify the period, vertical asymptotes, y-intercept, then sketch one period of the graph. * = #$% 4 +2 Find the period Period is ) 0 = ) ) = 1 Find the asymptotes = 0± ) 0 = 0± ) ) = ± The x-coordinate of the y-intercept is located halfway between ± and the y coordinate is the vertical shift. (0, 2)

9 Example Identify the period, vertical asymptotes, y-intercept, then sketch one period of the graph. * = 5#$% 3 Find the period Period is ) 0 = ) = 4 Find the asymptotes = 0± ) 0 = 0± ) = ±) The x-coordinate of the y-intercept is located halfway between ± ) and the y coordinate is the vertical shift. (0, -3)

10 Reciprocal Trigonometric Functions

11 Summary of Basic Trigonometric Functions

12 Symmetry Review Odd symmetry: Symmetric across the origin f(-x) = - f(x) Even symmetry: Symmetric across the y-axis f(-x) = f(x)

13 Symmetry in the unit circle shows that for any real number, the points P() and P(-), where x = cos and y = sin, located on the terminal side of an angle will have the same cosine values and opposite sine values. Thus, cos( ) = cos(θ)making cosine even and sin = sin () making sine odd.

14 Examples Using symmetry, find exact values of the following. SIN ( 4 3 ) Sine is an odd function, thus sin = sin () COS ( 4 3 ) Cosine is an even function, thus cos = cos () sin ) = sin ) = cos ) = cos ) =

15 Examples Using symmetry, find exact values of the following. SIN ( 4 4 ) Sine is an odd function, thus sin = sin () COS ( 4 4 ) Cosine is an even function, thus cos = cos () sin ) A = sin ) A = cos ) A = cos ) A =

16 Periodicity A function, f, is said to be periodic if there is a positive number P such that f( + P) = f() for all in the domain. The smallest number P for which this occurs is called the period of f. Sine and cosine are periodic functions and have a period of 24. Any point (x, y) on the unit circle, will be repeated after a rotation of ±24. Therefore, (x ±24,y±24) will be mapped on to (x, y). sin( ±2%4) = sin () $%Dcos ±2%4 = cos (), where n is any integer. In other words, you can add any multiples of 2πon to any angle and it would be the same as angle.

17 Use periodicity to evaluate the following: SIN (94) It is obvious that the angle 94 is more than one rotation of the unit circle. It is in fact 4 full rotations and 4 more. sin 94 = sin ( ) sin 94 = sin π = 0!" One complete rotation of the unit circle in terms of sixths would be 24 = ). G ) is at least twice ( A) ) around the unit G G) G circle, but not three times around it.!" ) G!" ) G = I G = cos ( H) G ) = cos ( H) G )

18 Use periodicity to evaluate the following: SIN ( ) One complete rotation of the unit circle in terms of thirds would be 24 = G). J) is at least four times unit circle, but not five sin J) sin J) = I K) = sin ( A) ) = sin A) A) around the times around it.!" One complete rotation of the unit circle in terms of fourths would be 24 = J) A. ) is at least once around the unit circle, A G) but not two times around it.!" ) A!" ) G = I A = cos ( ) A ) = cos ( ) A )