Section 3.8 Solving Trigonometric Equations and HC © Copyright all rights reserved to Homework depot: www.BCMath.ca

I) Review: Solving “Solving” means finding a value for the variable, so that both sides of an equation will be equal There are several ways to solve a Trigonometric equation: Algebraically: (Ch5.2) Isolate the variable using BEDMAS Use inverse trig. functions to find the solution(s) There are usually two solutions within one cycle (CAST) Use the reference angles to find the solution in standard position Graphically: Make Y1 equal to one side of the function Make Y2 equal to the other side Then find the points of intersection(s) Points where the graph intersect  the x-coordinates are the solutions © Copyright all rights reserved to Homework depot: www.BCMath.ca bcmath.ca

Ex: Solve the Equation for 0 ≤ θ ≤ 3π Make the left side of the equation y1 Make the right side of the equation y2 Solve the system by finding the points of intersection x y p 2p 3p -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Press: 2nd TRACE #5: Intersect 1st Curve? ENTER 2nd Curve? ENTER Guess? ENTER Repeat this process until you find all the points of intersection © Copyright all rights reserved to Homework depot: www.BCMath.ca bcmath.ca

Practice: Solve the Equation for 0 ≤ x ≤ 2π y -2p -p p 2p 1 2 3 4 5 6 7 © Copyright all rights reserved to Homework depot: www.BCMath.ca bcmath.ca

II) Finding the General solution: When finding the general solution, only one pair of solutions within a cycle is needed Other solutions can be founded by adding/subtracting multiples of the period (Co-terminal angles) Graphically, the horizontal distance between 2 solutions in separate cycles is equal to the period If you want to find solutions within other cycles, then just add/subtract the period x y p 2p 3p -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 The two general solutions will be: © Copyright all rights reserved to Homework depot: www.BCMath.ca bcmath.ca

III) Solving Trigonometric Functions with Horizontal Compressions: Ex: Find the value of θ, for 0 ≤ θ ≤ 360° The sine function is compressed horizontally by a factor of 3 The number of solutions will increase from 2 to 6 solutions between 0° and 360° x y p 2p -1 1 © Copyright all rights reserved to Homework depot: www.BCMath.ca bcmath.ca

Ex: Given the following system, how many solutions will there be between 0 ≤ θ ≤ 360° ,where “B” is a constant The constant “B” will indicate the horizontal factor of expansion/compression If B=2, then the graph will compress horizontally by a factor of ½. There will be 2 cycles between 0 & 360° x y 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 -1 If B=3, then the graph will compress horizontally by a factor of 1/3. There will be 3 cycles between 0 & 360° x y 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 -1 The number of solutions will be 2 x B © Copyright all rights reserved to Homework depot: www.BCMath.ca bcmath.ca

iv) Solving Trigonometric Equations Algebraically Isolate the variable using BEDMAS Use inverse trig. functions to find the solution(s) There are usually two solutions within one cycle (CAST) Use the reference angles to find the solution in standard position When there is a horizontal compression, add the period to find other sets of solutions © Copyright all rights reserved to Homework depot: www.BCMath.ca

v) Solving Equations with Horizontal Compressions Ex: Solve for “x”, 0 ≤ x ≤ 2π Make the substitution: 3x = θ Isolate the trigonometric Function Solve for θ in Quadrant 2 To Solve for “x”, substitute 3x back into θ Add the period to find solutions in the next cycle © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Solve for “x”, 0 ≤ x ≤ 2π Make the substitution: 2x = θ Isolate the trigonometric Function Solve for θ in Quadrant 4 To Solve for “x”, substitute 2x back into θ Add the period to find solutions in the next cycle © Copyright all rights reserved to Homework depot: www.BCMath.ca

VI) Solving Trigonometric equations Using Algebra Ex: Solve for “x”, 0 ≤ x ≤ 2π Factor © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Solve for “x”, 0 ≤ x ≤ 2π Factor © Copyright all rights reserved to Homework depot: www.BCMath.ca

VII) Solving Trigonometric equations Involving Substitution Ex: Solve for “x”, 0 ≤ x ≤ 2π Make the substitution: sin x = T Substitute T back into sin x © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Solve for “x”, 0 ≤ x ≤ 2π © Copyright all rights reserved to Homework depot: www.BCMath.ca

Viii) Using Double Angle Identities to Solve Equations Ex: Solve the following equation for 0 ≤ θ ≤ 360° Note: You are “Solving for θ”, not proving an identity!! Use the Double-Angle Identity Factor the equation Solve for θ from each equation. Remember to use CAST and there are TWO cycles between 0 and 360° Add the period (180°) to find other solutions © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Solve for 0 ≤ θ ≤ 360° Use the Double-Angle Identity Factor: Difference of Square Solve for θ from each equation. Remember to use CAST Note: There is only ONE cycle Between 0 and 360° © Copyright all rights reserved to Homework depot: www.BCMath.ca

Challenge: Solve for “x”, 0 ≤ x ≤ 2π Make the substitution: tan(2x) = T Add the period to find solutions in the next cycle © Copyright all rights reserved to Homework depot: www.BCMath.ca

Check: x y 0.5p p 1.5p 2p © Copyright all rights reserved to Homework depot: www.BCMath.ca

Homework Assignment 3.8