Slide 9- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Trigonometric Identities, Inverse Functions, and Equations Chapter 9

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.1 Identities: Pythagorean and Sum and Difference  State the Pythagorean identities.  Simplify and manipulate expressions containing trigonometric expressions.  Use the sum and difference identities to find function values.

Slide 9- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Basic Identities An identity is an equation that is true for all possible replacements of the variables.

Slide 9- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Pythagorean Identities

Slide 9- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply and simplify: a) Solution:

Slide 9- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued b) Factor and simplify: Solution:

Slide 9- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Simplify the following trigonometric expression: Solution:

Slide 9- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sum and Difference Identities There are six identities here, half of them obtained by using the signs shown in color.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find sin 75  exactly.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.2 Identities: Cofunction, Double-Angle, and Half-Angle  Use cofunction identities to derive other identities.  Use the double-angle identities to find function values of twice an angle when one function value is known for that angle.  Use the half-angle identities to find function values of half an angle when one function value is known for that angle.  Simplify trigonometric expressions using the double- angle and half-angle identities.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Cofunction Identities Cofunction Identities for the Sine and Cosine

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find an identity for Solution:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Double-Angle Identities

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find an equivalent expression for cos 3x. Solution:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Half-Angle Identities

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find sin (  /8) exactly. Solution:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Simplify. Solution:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.3 Proving Trigonometric Identities  Prove identities using other identities.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Logic of Proving Identities Method 1: Start with either the left or the right side of the equation and obtain the other side. Method 2: Work with each side separately until you obtain the same expression.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Hints for Proving Identities Use method 1 or 2. Work with the more complex side first. Carry out any algebraic manipulations, such as adding, subtracting, multiplying, or factoring. Multiplying by 1 can be helpful when rational expressions are involved. Converting all expressions to sines and cosines is often helpful. Try something! Put your pencil to work and get involved. You will be amazed at how often this leads to success.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Prove the identity. Solution: Start with the left side.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Prove the identity: Solution: Start with the right side. Solution continued

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Prove the identity. Solution: Start with the left side.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.4 Inverses of the Trigonometric Functions  Find values of the inverse trigonometric functions.  Simplify expressions such as sin (sin –1 x) and sin –1 (sin x).  Simplify expressions involving compositions such as sin (cos –1 ) without using a calculator.  Simplify expressions such as sin arctan (a/b) by making a drawing and reading off appropriate ratios.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Inverse Trigonometric Functions [0,  ][  1, 1] RangeDomainFunction

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find each of the following: a) b) c) Solution: a) Find  such that  would represent a 60° or 120° angle.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued b) Find  such that  would represent a 30° reference angle in the 2 nd and 3 rd quadrants. Therefore,  = 150° or 210° c) Find  such that This means that the sine and cosine of  must be opposites. Therefore,  must be 135° and 315°.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Domains and Ranges

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Composition of Trigonometric Functions

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Examples Simplify: Since  1/2 is in the domain of sin –1, Simplify: Since is not in the domain of cos –1,

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Special Cases

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Examples Simplify: Since  /2 is in the range of sin –1, Simplify: Since  /3 is in the range of tan –1,

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley More Examples Simplify: Solution: Simplify: Solution:

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.5 Solving Trigonometric Equations  Solve trigonometric equations.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Trigonometric Equations Trigonometric Equation—an equation that contains a trigonometric expression with a variable. To solve a trigonometric equation, find all values of the variable that make the equation true.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve 2 sin x  1 = 0. Solution: First, solve for sin x on the unit circle. The values  /6 and 5  /6 plus any multiple of 2  will satisfy the equation. Thus the solutions are where k is any integer.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphical Solution We can use either the Intersect method or the Zero method to solve trigonometric equations. We graph the equations y 1 = 2 sin x  1 and y 2 = 0.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another Example Solve 2 cos 2 x  1 = 0. Solution: First, solve for cos x on the unit circle.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphical Solution Solve 2 cos 2 x  1 = 0. One graphical solution shown.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley One More Example Solve 2 cos x + sec x = 0 Solution: Since neither factor of the equation can equal zero, the equation has no solution.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphical Solution 2 cos x + sec x

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Last Example Solve 2 sin 2 x + 3sin x + 1 = 0. Solution: First solve for sin x on the unit circle.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Last Example continued where k is any integer. One Graphical Solution

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.5 Rotation of Axes  Use rotation of axes to graph conic sections.  Use the discriminant to determine the type of conic represented by a given equation.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rotation of Axes When B is nonzero, the graph of is a conic section with an axis that is not parallel to the x– or y– axis. Rotating the axes through a positive angle  yields an coordinate system.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rotation of Axes Formulas If the x– and y– axes are rotated about the origin through a positive acute angle  then the coordinates of and of a point P in the and coordinates system are related by the following formulas.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Suppose that the axes are rotated through an angle of 45º. Write the equation in the coordinate system. Solution: Substitute 45º for  in the formulas

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Now, substitute these into the equation This is the equation of a hyperbola in the coordinate system.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Eliminating the xy- Term To eliminate the xy- term from the equation select an angle  such that and use the rotation of axes formulas.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph the equation Solution: We have

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Substitute these into and simplify. Parabola with vertex at (0, 0) in the coordinate system and axis of symmetry

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Discriminant The expression is, except in degenerate cases, is the discriminant of the equation The graph of the the equation 1. an ellipse or circle if 2. a hyperbola if and 3. a parabola if

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph the equation so Solution: We have The discriminant is negative so it’s a circle or ellipse. Determine  Substitute into the rotation formulas

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued After substituting into the equation and simplifying we have:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued with vertices and is an ellipse,

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.6 Polar Equations of Conics  Graph polar equations of conics.  Convert from polar to rectangular equations of conics.  Find polar equations of conics.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley An Alternative Definition of Conics Let L be a fixed line (the directrix); let F be a fixed point (the focus), not on L; and let e be a positive constant (the eccentricity). A conic is the set of all points P in the plane such that where PF is the distance from P to F and PL is the distance from P to L. The conic is a parabola if e = 1, an ellipse if e 1.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polar Equations of Conics To derive equations position focus F at the pole, and the directrix L either perpendicular or parallel to the polar axis. In this figure L is perpendicular to the polar axis and p units to the right of the pole.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polar Equations of Conics For an ellipse and a hyperbola, the eccentricity e is given by e = c/a, where c is the distance from the center to a focus and a is the distance from the center to a vertex. Simplified:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Describe and graph the conic Solution: Since e < 1, it’s an ellipse with vertical directrix 6 units to the right of the pole. The major axis lies along the polar axis. Let  = 0, and π to find vertices: (2, 0) and (6, π ). The center is (2, π ). The major axis = 8, so a = 4. Since e = c/a, then 0.5 = c/4, so c = 2 The length of the minor axis is given by b: b =

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Sketch the graph

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polar Equations of Conics A polar equation of any of the four forms is a conic section. The conic is a parabola if e = 1, an ellipse if 0 1.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polar Equations of Conics

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Converting from Polar to Rectangular Equations Use the relationships between polar and rectangular coordinates. Remember:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Convert to a rectangular equation: Solution: We have This is the equation of a parabola.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Finding Polar Equations of Conics We can find the polar equation of a conic with a focus at the pole if we know its eccentricity and the equation of the directrix.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find a polar equation of a conic with focus at the pole, eccentricity 1/3 and directrix Solution: Choose an equation for a directrix that is a horizontal line above the polar axis and substitute e = 1/3 and p = 2

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.7 Parametric Equations  Graph parametric equations.  Determine an equivalent rectangular equation for parametric equations.  Determine parametric equations for a rectangular equation.  Determine the location of a moving object at a specific time.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphing Parametric Equations We have graphed plane curves that are composed of sets of ordered pairs (x, y) in the rectangular coordinate plane. Now we discuss a way to represent plane curves in which x and y are functions of a third variable t. One method will be to construct a table in which we choose values of t and then determine the values of x and y.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Graph the curve represented by the equations The rectangular equation is

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parametric Equations If f and g are continuous functions of t on an interval I, then the set of ordered pairs (x, y) such that x = f(t) and y = g(t) is a plane curve. The equations x = f(t) and y = g(t) are parametric equations for the curve. The variable t is the parameter.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determining a Rectangular Equation for Given Parametric Equations Solve either equation for t. Then substitute that value of t into the other equation. Calculate the restrictions on the variables x and y based on the restrictions on t.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find a rectangular equation equivalent to Solution The rectangular equation is:

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determining Parametric Equations for a Given Rectangular Equation Many sets of parametric equations can represent the same plane curve. In fact, there are infinitely many such equations. The most simple case is to let either x (or y ) equal t and then determine y (or x ).

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find three sets of parametric equations for the parabola Solution

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Applications The motion of an object that is propelled upward can be described with parametric equations. Such motion is called projectile motion. It can be shown that, neglecting air resistance, the following equations describe the path of a projectile propelled upward at an angle  with the horizontal from a height h, in feet, at an initial speed v 0, in feet per second: We can use these equations to determine the location of the object at time t, in seconds.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example A baseball is thrown from a height of 6 ft with an initial speed of 100 ft/sec at an angle of 45º with the horizontal. a) Find parametric equations that give the position of the ball at time t, in seconds. b) Find the height of the ball after 1 sec, 2 sec and 3 sec. c) Determine how long the ball is in the air. d) Determine the horizontal distance that the ball travels. e) Find the maximum height of the ball.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Solution a) b)The height of the ball at time t is represented by y.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Solution c) The ball hits the ground when y = 0. The ball is in the air for about 4.5 sec.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example continued Solution d) Substitute t = 4.5 So the horizontal distance the ball travels is ft. e)Find the maximum value of y (vertex). So the maximum height is about 84.1 ft.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Applications The path of a fixed point on the circumference of a circle as it rolls along a line is called a cycloid. For example, a point on the rim of a bicycle wheel traces a cycloid curve.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Applications The parametric equations of a cycloid are where a is the radius of the circle that traces the curve and t is in radian measure.

Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example The graph of the cycloid described by the parametric equations is shown below.