Solving Trig Equations Objective: Solve many different Trig equations.

Introduction We are going to use standard Algebra techniques to solve a trig equation. Your primary goal in solving a trig equation is to isolate the trig function. For example, given, we would divide both sides by 2 and have

Introduction We are going to use standard Algebra techniques to solve a trig equation. Your primary goal in solving a trig equation is to isolate the trig function. For example, given, we would divide both sides by 2 and have We can use the unit circle and find that the in two places, x =  /6 and x = 5  /6. Since we aren’t working with inverse functions, we have no domain restriction.

Introduction These are not the only two places where this is true. Look at the graph below. There are an infinite number of places that the We express all of these answers by writing

Introduction We can also look at these answers as coterminal angles. We can move around the circle an infinite number of times and land on  /6 and 5  /6.

Example 1 Solve

Example 1 Solve Add the sinx to both sides

Example 1 Solve Add the sinx to both sides Subtract from both sides

Example 1 Solve Add the sinx to both sides Subtract from both sides Divide both sides by 2

Example 1 Using the reference angle of 45 0 or  /4, the in the third and fourth quadrant. The answers are

Example 2 Solve

Example 2 Solve Add 1 to both sides Divide both sides by 3 Take the square root of both sides

Example 2 Using the unit circle and a reference angle of  /6, the answers are in all four quadrants.

Example 3 Solve

Example 3 Solve Subtract 2cotx from both sides Take out the common factor Set each term equal to zero and solve

Example 3 Again, use the unit circle to solve. The cotx = 0 where cosx = 0.

Example 3 Again, use the unit circle to solve. The cotx = 0 where cosx = 0. There are no solutions to this equation since the range of cosx is [-1,1]

Example 4 Solve by factoring

Example 4 Solve by factoring Look at this as the quadratic function

Example 4 Solve by factoring for [0,2  ) Look at this as the quadratic function

Example 4 Now use these answers to solve for the sinx for [0,2  )

Example 4 Now use these answers to solve for the sinx for [0,2  )

Class Work Page 558 2, 8, 10, 12, 22, 26

Homework Page 558 1, 5-15 odd 21, 23, 25