Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 5: Section: Trigonometric Equations 5.1: Solving Conditional Equations I We have solved equations such as: cos x 3 for 0 x 2 or sin 1 for 90 90 2 What if we expanded the intervals above? (0 < x < 2π or 0º < θ < 360º) What if we place no restriction on the variable? (x any real number) Solve: cos x Solve: sin θ = ½ 3 for 0 x 2 2 for and for x R 0 360 and A Solve: tan 3 Solve: sin x = 0.2 Solve: tan x = -4.6 Solve: sec x =1.5 0 180 for for x when for x when for 0 x 360 0 x 360 0 < x < 2π and and A x A x A Approx to 4 decimal places. Explain why the following have no solutions: a) cos x = 3 b) csc x = -½ HW 5.1, pp295-296: 1 – 19 odd. and Section 5.2: Solving Trigonometric Conditional Equations II cos x 3 for 0 x 2 The equations we have solved so far have been simple equations in immediately solvable form in that they can be solved by inspection (recognizing a common angle from its functional value or by using inverse functions). The equations in this section require more algebraic manipulation to write in immediately solvable form. Appendix A has a review on solving algebraic equations. First Degree (Linear) for (0 < x < 2π) 2 cos x + 1 = 3 4 + 3 tan x = 2 tan x + 5 Second Degree (Quadratic) sin2 x – sin x – 2 = 0 (0 < x < 2π) cos2 t – 3 cos t = 2 (0 < x < 2π) tan x cos x = tan x (0 < x < 2π) 3 tan2 A = 1 for exact values of A. Find all solutions for cos2 t – 3 cos t = 2 HW 5.2, pp302-226: 1 – 43 odd, 61. 0° < A < 360° for t a real number Section 5.3: More Trigonometric Equations, Multi-Angle Equations In this section we will solve trigonometric equations that require both trig identities and algebraic manipulation. 3 cos x – 2 sin2 x = 0 (0 < x < 2π) 6 cos θ + 7 tan θ = sec θ 0° < θ < 360° sin t = cos t + 1 0 < t < 2π for sin 2x 3 for 0 x 360 2 sin 2x 3sin x 0 for 0 x 360 cos 4x cos x + sin 4x sin x = 0 0° < x < 360° cos2 2θ – 2 sin 2θ + 2 = 0 (0 < θ < 2π) Solving Trigonometric Equations 1. If the equation involves different multiples of the angle, use identities to obtain the same multiple. 2. If the equation involves more than one function, if possible, use identities to reduce it to involve only one function. If not, try the following: a) Square both sides of the equation. b) Multiply both sides by the common denominator to eliminate fractions. Then, if possible, use identities to rewrite the equation in terms of one function. 3. Identify the equation as linear or quadratic in form or as an equation that is factorable. Then use established algebraic procedures. HW 5.3, pp311-312: 1 – 31 odd, 43 – 55 odd. Section 5.4: Parametric Equations In algebra, graphs in the coordinate plane are usually represented by an equation in the variables x, and y. These equations are called Cartesian or rectangular. Ex: y = 2x2 + 3x +1. In Chapter 1, we introduced the unit circle x2 + y2 = 1 and then defined the x and y coordinates in terms of arc length t. x = cos t and y = sin t We have introduced a third variable t We now define the variable t as a parameter and the equations x = f (t) = cos t and y = g (t) = sin t as parametric equations A parameter can represent time, arc length or an angle. In Chapter 1 we parametrically defined the x- and y-coordinates on the unit circle as continuous functions of t. x = cos t and y = sin t Where x2 + y2 = 1 Ex: Let’s let t be units of time (0 < t < 2π) for an object travelling along the unit circle a) Where is the object at t = 0 ?, t = π/2 ? b) At what time will the object be at the point (0, -1)? c) What is the direction of motion? d) How long will it take object to return to its starting position? t x = cos t y = sin t 0 π/2 π 3 π/2 2π Use an arrow on the curve to indicate direction (orientation) of travel. Ex: Describe the motion of x = sin 2t and y = cos 2t, where t represents time and 0 < t < π, by finding the rectangular equation for the curve. Then graph the motion indicating the starting and ending points, the orientation and then find the time it takes to return to the starting position. cos2 2t + sin2 2t =1 y2 + x2 = 1 or x2 + y2 = 1 t x = sin 2t y = cos 2t 0 π/4 π/2 3 π/4 π Steps for Sketching Parametric Equations 1. Eliminate the parameter (if possible) to find the rectangular equation. If x and y are defined in terms of sine and/or cosine, try to use the Pythagorean identity to eliminate. For other trig functions, try to use an equivalent form of the Pythagorean identity. If x and y are defined in other terms, try the method of substitution. 2. Set up a table of t x, and y, with increasing (smallest to largest) values of t. 3. Plot several points on the curve in order of the increasing values of the parameter. Ex: A particle moves in the plane with its x- and y-coordinates varying with t according to x = 2cos t and y = 3sin t, and 0 < t < 2π. 1. Eliminate the parameter to find the rectangular equation. 2. Find the starting and ending points of motion on the indicated interval of t. 3. Sketch the graph that describes the motion and use an arrow to indicate the orientation (direction of increasing values of t). t x = 2 cos t y = 3 sin t 0 π/2 π 3π/2 2π HW 5.4, pp320-321: 1 – 27, odd.