Trig Equations © Christine Crisp AS Use of Maths

Trig Equations or To solve trig equations you have to know what the sine and cosine curves look like Due to the symmetrical appearance of the graphs when solving trig equations there will be more than one answer Y=sinx Y=cosx

Trig Equations Y=sinx Y=cosx Ex Sin 45 = 0.7 And Sin 135 = Ex Cos 60 = 0.5 And Cos 300 =

Trig Equations To solve trig equations use the forwards and backwards method Solve the equation This means that if you find the sin of x then the answer is 0.5 This is pronounced inverse sin x and is on the same key as sin x but in yellow so use the 2 nd F key The opposite or inverse of sin x is sin –1 x Remember an inverse function is a function which has the opposite effect The inverse (opposite) of x 2 is  x

Trig Equations Solve the equation Forwards x  sin it  = 0.5 Backwards 0.5  sin -1 it  x x = sin = 30 o So the solution to the equation sinx = 0.5 is x = 30 o But unlike normal algebraic equations trig equations have many answers because the trig graph is periodic and repeats every 360 o

Trig Equations BUT, by considering the graphs of and, we can see that there are many more solutions: e.g.1 Solve the equation. Solution: The calculator gives us the solution x = Every point of intersection of and gives a solution ! In the interval shown there are 10 solutions, but in total there are an infinite number. The calculator value is called the principal solution principal solution

Trig Equations We will adapt the question to: Solution: The first answer comes from the calculator: Use the sin -1 key Solve the equation for Forwards x  sin it  = 0.5 Backwards 0.5  sin -1 it  x x = sin = 30 o This limits the number of solutions

Trig Equations 1 Add the line Sketch between There are 2 solutions. The symmetry of the graph shows the 2 nd solution is It’s important to show the scale. Tip: Check that the solution from the calculator looks reasonable.

Trig Equations Solution: The first answer from the calculator is e.g. 2 Solve the equation in the interval Forwards x  cos it  = -0.5 Backwards -0.5  cos -1 it  x x = cos = 120 o The opposite or inverse of cos x is cos –1 x (inverse cos x)

Trig Equations 1 Solution: The first answer from the calculator is Add the line e.g. 2 Solve the equation in the interval Sketch between There are 2 solutions. The symmetry of the graph shows the 2 nd solution is

Trig EquationsSUMMARY Find the principal solution from a calculator. Find the 2 nd solution using symmetry where c is a constant  To solve orfor or Draw the line y = c. Sketch one complete cycle of the trig function. For example sketch from to.

Trig EquationsExercises 1.Solve the equations (a) and (b) for Forwards x  cos it  = 0.5 Backwards 0.5  cos -1 it  x x = cos = 60 o

Trig Equations 1 Exercises The 2 nd solution is 1.Solve the equations (a) and (b) for Solution: (a) ( from calculator )

Trig Equations (b) Exercises Forwards x  sin it  = Backwards  sin -1 it  x x = sin -1 = 60 o

Trig Equations 1 Solution: ( from calculator ) The 2 nd solution is (b) Exercises

Trig Equations 1 e.g. 5 Solve the equation for Since the period of the graph is this solution is Solution: More Examples Using forwards and back

Trig Equations 1 Solution: e.g. 5 Solve the equation for Symmetry gives the 2 nd value for. The values in the interval are and More Examples

Trig Equations 1 Solution: Principal value e.g. 6 Solve for By symmetry, Method Ans: Using forwards and back

Trig EquationsSUMMARY  To solve or Once 2 adjacent solutions have been found, add or subtract to find any others in the required interval. Find the principal value from the calculator. Sketch the graph of the trig function showing at least one complete cycle and including the principal value. Find a 2 nd solution using the graph.

Trig Equations 1.Solve the equations ( giving answers correct to the nearest whole degree ) (b) for (a) for Exercises

Trig Equations 1 (a) for Solution: Principal value By symmetry, Ans: Exercises Using forwards and back

Trig Equations 1 Ans: (b) for Solution: Principal value Exercises Using forwards and back

Trig Equations Solve the following (b) Sinx = 0.49 for (a) Sinx = 0.83 for (d) Cosx = 0.65 for (c) Cosx = 0.25 for Answers a) 56.2 o, 123.9b) 29.3 o, b) 75.5 o, 284.5c) 49.5 o, 310.5

Trig Equations