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This 19 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1330 at University of Houston taught by Mary Flagg in Fall. Since its upload, it has received 14 views. For similar materials see /class/208405/math-1330-university-of-houston in Mathmatics at University of Houston.
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PreCalSection53 Page 1 of7 Math 1330 Section 53 Graphing the Other Trig Functions Since the tangent cotangent secant and cosecant are de ned as fractions they all have points where the functions are unde ned In all cases these X values are vertical asymptotes for the graphs of the functions Tangent sinx Remember that tanx so tangent 1s undefined for values of X where cos1ne 1s cos x 7239 37239 57239 zero S1nce cosx 0 when x ii i tangent has vert1cal asymptotes at 77 the lines x at for all integers k The period of the tangent functlon 1s 7239 So one perlod 1s cons1dered to run from E to 7239 2 We can use the values of tangent for the special angles to help construct the graph Tangent Values Graph of basic cycle Extending the graph for more than one cycle PreCalSection53 Page 2 of7 Cotangent cosx sinx asymptote lines at the lines x kn39 for all integer values of k Since cotx cotangent is unde ned where sine is zero This means cotX has Period of the cotangent function is 7239 One period is considered to run from 0 to 7239 Use the values for cotangent of the special angles to draw the graph Cotangent Values Graph of basic cycle Extend Graph Secant Since secant is de ned as the reciprocal of the cosine function secant is unde ned where cos1ne 1s zero Therefor secX has vert1cal asymptotes at x 3 kn39 The peiod for the secant function is the same as for cosine 27239 The range ofthe cosecant function is 00 1 U 1 oo PreCalSection53 Page 3 of7 We can graph cosecant by starting with the graph of cosine here s how Cosecant Since cosecant is de ned as the reciprocal of the sine function it is unde ned Where sine is zero This means cscX has asymptote lines at the lines x at for all integer values of k The period of cosecant is 27239 the same as the sine function The range of cscX is oo lu 1oo One cycle is usually considered to be the interval 0 27239 We can construct the graph of the cosecant function by starting with the sine function here s how PreCalSection53 Page 4 of7 Transformations with tan and cot How to graph y AtanBx C or y AcotBx C Find the new asymptote lines State the interval for one period between the asymptotes and state the period Divide the period into quarters Find the Xintercept which is the center of the period Find the 1A and 3 points and their yvalues connect with a smooth line that approaches asymptotes 99559 Step 1 Finding the new asymptote lines Tangent For tan set Bx C and solve for X for the asymptote line on the left end of the shifted basic period and Bx C 3 and solve for X for the equation of the asymptote line on the right side of the shifted basic period Draw the new asymptote lines Step 2 State the period The period iszg The interval for one basic period is ab where Xa and Xb are the asymptote lines found in step 1 Step 3 Divide the basic period into quarters by first marking the middle halfway point and then the quarter points The middle point is the Xintercept The quarter points are the tantA or 7A points in the basic shape of tangent Step 4 If A is negative the graph must be re ected across the XaXis Example 1 Graph over one period y 2tan2x g PreCalSection53 Page 5 of7 Cotangent The procedure is the same as for tangent but the new asymptote lines that are the edges of the basic period are found by setting Bx C0 and Bx C7r Example 2 Graph over one period y 2 cot7239x Secant and Cosecant Graphing the transformed functions for AsecBx C and A cscBx C are most easily done by starting with the graphs of AcosBx C or AsinBx C It is easiest to do this by example Example 3 Graph the function 2sec PreCalSecti0n53 Page 6 of7 Example 4 Graph the function y csc3x Example 5 Graph the function y 2 sec27rx 7239 PreCalSecti0n53 Page 7 of7 Example 6 Finding Asymptotes Find 3 asymptotes for the functions listed fx 3sec2x fx 4tan7rx fx cot 27m x fx csc Example 7 Find Xintercepts for the functions listed 2 cos4x sec2x 7239 tan x l 3 7m 7239 s1n 2 4 PreCalSection4 4 Math 1330 Section 44 Trig Expressions and Identities Algebraic Operations with Trig Functions We can manipulate expressions with trig functions using the same techniques we use when manipulating polynomials or rational functions These techniques include distributing collecting like terms putting everything over a common denominator and factoring Example 1 Multiply sint9 3 sint9 6 5 cost9 sint9 Example 2 Combine Example 3 Simplify 9cost9 tant9 7cost9tant9 Example 4 Factor 16 cos2 9 81sin2 9 Exam le 5 Factor tan2 9 8tan 9 12 P cos2 9 cost9 12 Example 6 Factor cost9 3 PreCalSection4 4 2 Identities An identity is an equation with a variable that is true for all values of that variable The trig functions are de ned in such a way that there are many identities involving trig functions I will list the basic identities here and we will encounter more identities as we work with trig functions It is necessary to memorize the basic identities that I will list It is not necessary to memorize other identities but you need to be able to use them and prove they are true Basic Identities from the Definitions Note that all of these ratios are undefined when the function in the denominator is 0 I will not list the separate if X is not 0 conditions in this list to help keep the list easy to remember Definition Identities 1 tan 9 sm 6 cos 9 2 cot t9 cf St9 sm 9 l 3 cot t9 tan 9 4 sec 9 cos 9 l 5 csct9 sm 9 Pythagorean Identities 6 sin2 t9cos2 9 1 7 tan2 61 sec2 9 8 lcot2 t9 csc2 9 Even 7 Odd Function Identities 9 sin t sint 10 cos t cost ll tan t tant 12 csc t csct l3 sec t sect l4 cot t cott The most important ones to memorize are l56910 The others can be derived directly from these identities PreCalSection4 4 Using Trig Identities to Evaluate Trig Functions Example 7 A cos 45 B tamg Example 8 If csct9 g and lt 9 lt 7239 nd the exact values for the other 5 trig functions for 9 Using Trig Identities to Simplify Expressions Example 9 Simplify tanx cscx csc2 t9 l Exam le 10 Sim lif p p y l sin2t9 sec2 9 Exam le 11 Sim lif p p y tant9cott9 PreCalSection4 4 4 Example 12 Simplify csct9 cott9sect9 l Proving Identities You prove an identity by showing how you can algebraically manipulate one or both sides of the equation so that they both are the same Your proof should look like Left side Right Side This is NOT solving an equation it is using the other identities and definitions to simplify one side of the equation so that it looks exactly like the other side or simplify both sides so they look exactly alike Helpful Hints l 2 9939er 7 8 Start with the messiest side Draw a line under the equals sign or put a question mark over it so you remember not to move terms from one side to the other like solving an equation Get common denominators Convert everything to sine and cosine this often helps but not always Use the Pythagorean identities especially sin2 9 cos2 9 1 If you see l cos 9 in a fraction try multiplying both top and bottom by l cost9 and using the Pythagorean identity The same goes for 17 or 17 with other trig identities Try multiplying top an bottom by the other sign If you get stuck working on one side try the other Try substituting l for sin2 t9cos2 9 or substitute sin2 t9cos2 9 for 1 Example 13 Prove the following identity secx sinx tanx cosx PreCalSection4 4 Example 14 Prove the following identity cotAl tan2 A tam1 csc A Example 15 Prove the following identity sinx 1 0080C cscx cotx Example 16 Prove the following identity csc4 x csc2 x cot4 x cot2 x Example 17 Prove the following identity 1 cosx cscx cotx sinx PreCalSection21 Page 1 of7 Math 1330 Section 21 Linear and Quadratic Functions A linear function is one that has the form y mxb Where in and b are constants The graph of a linear function is a line With slope in and yintercept b The sign of the slope indicates Whether the line is an increasing or a decreasing function mlt0 I ago v7 v L 1 07 L m 0 is a horizontal line in unde ned is a vertical line PreCalSecti0n2l Page 2 of7 Example 1 Graph the equation using its slope and intercept y 3x 2 Example 2 Graph f x 2x 3 Parallel Lines Two lines are parallel if they never cross In terms of equations two lines are parallel if they have the same slope Perpendicular Lines Two lines are perpendicular if they meet at a right angle In terms of equations two lines with slopes m1 and m2 are perpendicular if and only if m1 m2 l Formulas f0r Lines Slope Given points qy1 and x2 y2 the slope of the line passing yz yr x2 x1 through these points is m PointSlope Given a point qy1 and a slope m the equation for the line that passed through the point having slope m is y y1 mx x1 SlopeIntercept The slopeintercept equation for a line is an equation of the form y mx b where m is the slope and b is the yintercept PreCalSecti0n2l Page 3 of7 Writing Eguations of Lines Example 3 Write the equation of the linear function f such that f 2 1 and f3 2 Example 4 Write the equation of the linear function fx that passes through the point 45 and is parallel to the line 4x3 y 7 Example 5 Find the linear function f that is perpendicular to the line containing 42 and 104 and passes through the midpoint of the line segment connecting these points Example 6 Find the linear function f that passes through the point 36 and such that f 1 8 9 PreCalSecti0n2l Page 4 of7 Quadratic Functions Definition A quadratic function f is a function that can be put in the form fx ax2 bxc where a at 0 and a b and c are constants Graphing a Quadratic Function Standard Form fx ax h2 k The graph of f is a parabola The axis of symmetry is the vertical line xk The direction the parabola open depends on the sign of the constant a If a gt 0 then the parabola opens up If a lt 0 then the parabola opens down Graphing a quadratic equation procedure 1 Put the function into standard form by completing the square 2 Identify the vertex 3 Use the transformation rules to transform the graph of y x2 into the given function Maximum or Minimum Value of a Quadratic Function A quadratic function has either a maximum or minimum value depending on whether the parabola opens up or down This max or min is the Y VALUE of the vertex If the coefficient of the xsquared term is positive the parabola opens up and the vertex is a MINIMUM If the coefficient of the xsquared term is negative the parabola opens down and the vertex is a MAXIMUM PreCalSecti0n2l Page 5 of7 Example 7 Write the quadratic equation in standard form by completing the square Identify the vertex and the axis of symmetry and the maximum or minimum value Then sketch the graph fx 2x2 4xl fx 3x2 6x l PreCalSection21 Page 6 of7 A Formula for Finding the Vertex A quadratic function in the form f x ax h2 k Z Formula the Vertex is the Point f 1 61 Example 8 Let f x x2 3x3 Find the vertex and maximum or minimum Finding the Intercepts of the Graph of a Quadratic Function YIntercept is just the value of the function at x 0 The xintercepts are the values of x such that fx0 To find them we need to solve the equation fx0 Methods for solving f x axz bx c l factoring 2 completing the square 3 the quadratic formula The uadratic Formula The solutions to the equation f x ax2 bx c are the values b i I b2 4610 2a The expression 12 4ac is called the discriminant If b2 4616 gt 0 then the quadratic has 2 real roots two xintercepts x If b2 4610 lt 0 then the quadratic equation has no real roots it does NOT cross the x axis at all If b2 4616 0 then the quadratic has just one real root ie one xintercept The intercept is the vertex of the parabola which geometrically means the parabola sits on the x axis and either opens up or down PreCalSecti0n2l Page 7 of7 Example 9 For the following quadratic functions find a the vertex b the axis of symmetry the line x x value of the vertex 0 the maximum or minimum value 1 the yintercept e the xintercept t sketch the graph fx3x2 6x5 fx x2 8xl6
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