Solving Equations Pure Math : Explained! www.puremath.com 55
Part One - Graphically Solving Equations Solving trigonometric equations graphically: When a question asks you to solve a system of trigonometric equations, they are looking for the values of θ that make both equations true. There are two ways you can solve for θ : graphically in your TI-8, and algebraically. Part I will show the graphing method, and Parts II & III will focus on algebraic methods. Example : Solve cos θ = and state the general solutions: In your TI-8, graph each equation in degree mode. Now use nd Trace Intersect to find the points of intersection. They occur at 6º & º Example : Solve cos θ = If you extend the window, you will see that the intersection points are in the same relative places, one period later. The first general solution is: 6º ± n(6º) or ± n( ) and the second is: 5 º ± n(6º) or ± n( ) and state the general solutions: Graph both equations in your TI-8, then solve for the first two intersection points. The first two intersection points are at.5º and 57.5º. As you can see in the graph, the solutions repeat themselves every period. Since the b-value is, the period is 8º, or. The first general solution is:.5º ± n(8º) or ± n 8 And the second is: 57.5º ± n(8º) or 7 ± n 8 Pure Math : Explained! www.puremath.com 56
Part One - Graphically Solving Equations Example : Graphically find the general solutions for sinθ - = Graph the two equations in your TI-8 and solve by finding the points of intersection. 6º ± n(6º) or ± n( ) º ± n(6º) or ± n( ) (In this case, another method would be to find the x-intercepts using nd Trace Zero) Note that even if you manipulate the equation, you can still solve by graphing: If you re-arrange the equation to sin θ = by taking to the other side, we get: Solving, we still have the same answers of 6º & º If we manipulate the equation again by dividing both sides by, we get: Solving this: sin θ =. Once again, we still get the same answers of 6º & º Manipulating an equation does NOT change the solution! Pure Math : Explained! www.puremath.com 57
Part One - Graphically Solving Equations Find the general solution (In degrees & radian fractions) for each of the following equations: ) sin x = ) cos x = ) sin x = 4) sin 4x = 5) tan x = 6) sin x.5 = 7) tan x + = 8) sin x = Pure Math : Explained! www.puremath.com 58
Part One - Graphically Solving Equations ) ± n( ) ± n 9 4 ± n( ) ± n 9 ± n(8 ) ± n 6 6) 5 5 ± n(8 ) ± n 6 ) ± n(8 ) ± n 7) ± n(8 ) ± n ) n ± n(9 ) ± 8) 8 48 ± n(7 ) ± n 4 6 ± n(7 ) ± n 4 ( ) ( ) 4) 7 n 5.5 ± n(9 ) ± 4 n 8.5 ± n(9 ) ± 4 5) ± n(9 ) ± n 6 Pure Math : Explained! www.puremath.com 59
Part Two - Linear Equations In this lesson, we will algebraically solve trigonometric equations. There are two main types of equations you will be asked to solve: linear & nonlinear. Note that you will still be able to solve all of these in your calculator as you did in the previous lesson, but on the diploma they frequently have written response questions where you need to present an algebraic solution. Linear Trigonometric Equations: Example : Solve: 4cosx + = in the domain x< To solve this, we must get 4cosx + = 4cosx = - - cosx = 4 cosx = - cosx by itself on the left side of the equation. From the unit circle, we know co sx is 4 - when x = & Example : Solve: sinxcosx = sinx in the domain x< We need to bring everything over to the left side, then factor. sinxcosx - sinx = sinx(cosx -)= Now set each factor on the left side equal to zero: sinx = x=, We don t include since the domain is using a < sign, not a sign. cosx -= cosx = cosx = 5 x=, You may be tempted to divide both sides of the equation by sinx to cancel it out. Don t do this! In math, you are never allowed to cancel variables on opposite sides of the equation. Suppose you made this error and canceled sinx. Then you would get: sinxcosx = sinx cosx = cosx = 5 x=, As you can see, we lose solutions of & doing it this way. Pure Math : Explained! www.puremath.com 6
Part Two - Linear Equations Example : Solve: sinx sinx + = 4 in the domain x< sinx sinx + = 4 sinx += 4sinx = sinx x= Multiply both sides by the common denominator, which is. This will eliminate the fractions. Example 4: Solve sinxsecxcotx = sinxsecx in the domain x< sinxsecxcotx - sinxsecx = sinxsecx(cotx -)= Bring all terms to one side so you can factor. sinx = x=, secx = = cosx x=no solutions cotx -= cotx = 5 x=, The combined solution set is:,,, 5 Example 5: Solve s inx = in the domain x< sinx = sinx = Since is not on the unit circle, we are forced to do this equation graphically in the calculator. 4.8º =.7 rad 8.º =.4 rad Pure Math : Explained! www.puremath.com 6
Part Two - Linear Equations Solve each of the following in the domain x< ) sin x = ) cosx = ) 4sinx + = sin x + 4) sin x cosx= cosx 5) tanx = 6) cosx + = 7) sin xcos xtan x+ sin xcos x= 8) cos x(cos x+ ) = Pure Math : Explained! www.puremath.com 6
Part Two - Linear Equations 9) (sec x+ )(sin x )(cot x ) = ) csc xcot x+ cot x = ) tan xcos x+ cos x= ) (cos x )(tan x ) = ) sec xcos x+ sec x= 4) sin x sin x = 5) tan x tan x = 6) 6 csc x csc x 6 + = 5 5 Pure Math : Explained! www.puremath.com 6
) ) 5 x =, 6 6 x =, 6 6 Trigonometry Lesson Part Two - Linear Equations ) ) 5 x =,, sec x(cos x+ ) = x = ) x = sin x = sin x sin x sin x= 4) sin xcosx cosx= sin x = 4) cos x(sin x ) = x =, 5 x =,,, 6 6 tan x tan x 6 = 6 6 tan x = tan x tan x= 5) 5) 5 x =, tan x = 7 x =, cos x = 6) 5 x =, 6) sin xcos x(tan x+ ) = csc x csc x 6 5 + = 5 7) 7 5 5 x =,,,,, 4 4 csc x+ 5csc x= 6 8cscx = 6 4 8) x =,,, csc x = sin x = 9) 5, x =,, 5 4 4 x =, 6 6 ) cot x(csc x+ ) = x =, ) cos x(tan x+ ) = 7 x =,,, 4 4 Pure Math : Explained! www.puremath.com 64
Part Three - Nonlinear Equations Quadratic Trigonometric Equations: Example : Solve 4sin x = sin x = 4 4sin x - = in the domain x< sin x = 4 sinx = ± 4 5 x=,,, Example : Solve cos x - cosx = in the domain x< cos x - cosx = cosx(cosx -)= cosx = x=, cosx -= cosx = x= The complete solution is: x=,, Example : Solve cos x = cosx - in the domain x< cos x - cosx += (cosx -)(cosx -)= cosx -= cosx = cosx = 5 x=, cosx -= cosx = x= The complete solution is: 5 x=,, Pure Math : Explained! www.puremath.com 65
Part Three - Nonlinear Equations Example 4: Solve sin x -5sin x +6sinx = in the domain x< sinx(sin x - 5sinx +6)= sinx(sinx - )(sinx - )= sinx = x=, sinx - = sinx = x=no solution sinx - = sinx = x=no solution 8 4 Example 5: Solve tan x - tan x = in the domain x<, and state the general solution tan x(tan x -)= 4 tan x(tan x +)(tan x -)= 4 tan x(tan x +)(tanx +)(tanx -)= 4 tan x = 4 tan x = tanx = x=, tan x += tan x = - No solution The complete solution set for x < is: Watch out for difference of squares! tanx += tanx = - 7 x=, 5 7 x =,,,,, tanx -= tanx = 5 x=, The general solution is: x=±n and x= ±n 4 These two general solutions will account for all the angles we found. Example 6: Solve cot x - cotx -= in the domain x< This equation cannot be factored, so graph and find the solutions in radian decimals:.9 rad,.98 rad, 4.6 rad, 5. rad Graph in your TI-8 as: - - (tan(x)) tan(x) nd Then use Trace Zero to find x-intercepts. Pure Math : Explained! www.puremath.com 66
Part Three - Nonlinear Equations Solve each of the following in the domain x< ) cos x = 4 7) 6cos x cos x = ) sin x = 4 8) sin x sin x+ = ) tan x = 9) 4cos x+ cosx = 4) 4sin x = ) cos x+ cos x cos x= 5) sin x sin x = 4 ) tan x tan x= 6) sin x = sin x 8 4 ) cos x cos x= Pure Math : Explained! www.puremath.com 67
Part Three - Nonlinear Equations ANSWERS: 5 7 ) x =,,, 6 6 6 6 5 7 ) x =,,, 6 6 6 6 9) ( cos + cos ) = x ( cos x )(cos x+ ) = 5 x =,, x 5 7 ) x =,,, 4 5 4) x =,,, ) cos ( cos + cos ) = x x x cos x( cos x )(cos x+ ) = 5 x =,,,, 5) 6) (sin x + )(sin x ) = 7 x =,, 6 6 (sin x )(sin x + ) = 5 x =,, 6 6 ( cos cos ) = x x 7) ( cos x+ )(cos x ) = 4 x =,, ) x x tan (tan ) = x x+ x = tan (tan )(tan ) 5 7 x =,,,,, x x cos (cos ) = 4 cos x(cos x+ )(cos x ) = ) x x+ x+ x = 4 cos (cos )(cos )(cos ) x =,,, (sin x )(sin x ) = 8) 5 x =,, 6 6 Pure Math : Explained! www.puremath.com 68
Part Four - Algebraically Solving Multiple Angles Algebraically Solving Double, Triple, and Half Angles: While technology can be used to solve equations involving double & triple angles, it is advantageous to understand the algebraic process involved with these question types. Example : Solve sin θ = algebraically over the interval x. To solve this equation algebraically, you need to perform the following steps: Step ) Start by solving the equation sin x =. We can do this easily using the unit circle. The answer to this equation is x = and. Step ) Add to each of the angles we found. 6 7 + = + = 6 8 + = + = Don t forget that adding fractions requires a common denominator. 7 8 Step ) Finally, take all your solutions,,, and divide by. (Or multiply by ½) 7 4 The answer is,,, 6 6 The general solution is: You can verify the results by graphing and checking the x-coordinates of the points of intersection. Pure Math : Explained! www.puremath.com 69
Part Four - Algebraically Solving Multiple Angles Example : Solve cos θ = for the domain θ Step ) Start by solving the equation cos x = 7 The answer to this equation is x = and.. We can do this using the unit circle. Step ) Add to each of the angles we found. 8 9 + = + = 7 7 8 5 + = + = Step ) Add to each of the angles we found in Step. 9 9 8 7 + = + = 5 5 8 + = + = 7 9 5 7 Step 4) Finally, take all your solutions,,,,, and divide by. (Or multiply by /) 7 5 7 The answer is,,,,, The general solution is: You can verify the results by graphing and checking the x-coordinates of the points of intersection. Pure Math : Explained! www.puremath.com 7
Part Four - Algebraically Solving Multiple Angles Example : Solve sin θ = for the domain θ To solve this equation algebraically, you need to perform the following steps: Step ) Start by solving the equation sin x = The answer to this equation is x = and.. Do this using the unit circle. 4 Step ) Divide each angle by (Or multiply by ) to obtain and. The general solution is: You can verify the results by graphing and checking the x-coordinates of the points of intersection Questions: Algebraically solve for θ over the interval θ 4) sin θ = ) sin θ = ) ) co s θ = co s θ = 5) 6) co sθ = cos θ = Answers: 5 5 ) θ =,,, 6 6 ),, θ =, 4 5 ) θ =,,, 7 4) θ =, 5) 5 5 θ =,,,,, 8 8 8 8 8 8 6) θ = Pure Math : Explained! www.puremath.com 7