Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.

Transcription

1 Section Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve: x 1 7 (one solution!) Quadratic Equation: Solve x 4x 5 0 (two solutions!) Now, we will work on trigonometric equations. An equation that involves a trigonometric function is called a trigonometric equation. Example: Trigonometric Equation: Solve sin( x) 1. This means find all values of x satisfying the given information sin( x) 1. 1

2 Let s solve the equation: sin( x) 1. 1 st approach: Think about the unit circle; which angle has sine equal to 1 on the unit circle? Answer: x. So, over the interval 0, (one rotation around the unit circle), there is only one solution to this equation. Now, remember that sine is periodic with period, so the angles, 4, 6, etc. all have the same sine value. Solution set: k : where k is an integer (infinitely many solutions!)

3 nd approach: Graph f ( x) sin( x) and see when it intersects the line y 1. There is only one intersection over the interval 0, : x. If you graph the big picture, you will see many more intersections: Solution set: k : where k is an integer (infinitely many solutions!) 3

4 Example: Find all solutions to the equation: 1 sin( x) Brainstorming: We know that sin and sin So, the first angles that come to mind are: x and x. 6 6 That is, over one period, there are TWO solutions: x and 6 5 x. 6 To find ALL solutions: add multiples of the period two each solution: x, x, x 4, and so on x, x, x 4, and so on Therefore, the solutions of the equation are: x k 6, 5 x k 6, where k is any integer. Again, there are infinitely many solutions to this equation. 4

5 Recall: For sine and cosine functions, the period is. For tangent and cotangent functions, the period is. 5

6 Example: a) Solve the equation in the interval 0, : 1 cos( x) b) Find all solutions to the equation: 1 cos( x) 6

7 Example: a) Solve the equation in the interval 0, : tan( x) 1 b) Find all solutions to the equation: tan( x) 1 7

8 POPPER for Section 6.3: Question#1: Solve sin( ) x over the interval 0,. a) x b) x 4 3 c) x, x d) x, x e) x, x, x f) None of these 8

9 Remark: For more complicated equations, the first step in solving a trigonometric equation is to isolate the trig part. Equation: cos( x) 1 ; Isolate the cosine: 1 cos( x). Equation: sin( x) 5 6 ; Isolate the sine: sin( x) 1. Equation: sin( x) 4 5 ; Isolate the sine: 1 sin( x) 1sin( x). 9

10 Example: a) Solve the equation in the interval 0, : cos( x) 4 3 b) Find all solutions to the equation: cos( x)

11 Example: Solve the equation in the interval 0, : sin( x)

12 POPPER for Section 6.3: Question#: Solve 4cos( x) 1 3 over the interval 0,. 4 a) x, x 3 3 b) x 3 5 c) x, x d) x, x 3 3 e) x, x 3 3 f) None of these 1

13 Remark: If the trig expression is in the form sin( Bx ) or cos( Bx ), this changes the period. Here are graphs of f ( x) sin( x) and gx ( ) sin( x) over one period: Solve sin( x) 1 ; solution: x Solve sin( x) 1 ; then x solution: x 4 13

14 Example: a) Solve the equation in the interval [ 0, ) : sin( x) b) Find all solutions to this equation. Add multiples of the period to your solutions from part (a)! Example: Find all solutions to the equation: cos(4 x)

15 Example: Find all solutions to the equation: cos(4 x)

16 Example: a) Find all solutions of the equation in the interval [0,4 ): x sin 3 x b) Find all solutions of the equation in the interval0,8 : sin 3 16

17 Example: Solve the equation in the interval [0,): x cot 1 17

18 Example: Find all solutions of the equation in the interval 0, : cosx 4 18

19 Example: Find all solutions of the equation: 4sinx 1 19

20 Sometimes, the expressions will have power of trig functions. Again, the first step is to isolate the trig expression. In some cases, you will need to use factoring techniques you learned in algebra (see section 4.4 for some examples). Example: Solve the equation in the interval 0, : sin ( x) 1 3 0

21 Example: Solve the equation in the interval 0, : sin ( x) 5sin( x) 3 0 1

22 Example: Solve the equation in the interval 0, : 3 tan x tan x 0

23 Example: Solve the equation in the interval 0, : csc x 4 3

24 Example: Solve the equation in the interval 0, : sin ( x)cos( x) cos( x) 4

25 Example: Let f ( x) cos ( x) sin ( x). Find the x intercepts of this function over the interval 0,. 5

26 POPPER for Section 6.3: Question#3: Solve cos ( x) 6 7 over the interval 0,. a) x, x b) x 0, x c) x 0, x, x 3 d) x, x e) x0, x, x f) None of these 6

27 Now, let s practice finding the number of solutions to an equation without solving it. Note that the interval that you are using really matters here. How many solutions are there for the following equation on the interval 0,? a) sin( x) 1 Number of solutions: b) 1 sin( x) Number of solutions: 4 c) 1 sin( x) Number of solutions: 5 d) sin( x) 1 Number of solutions: e) sin( x) Number of solutions: 7

28 Example: How many solutions are there for the following equation on the interval 0,4? a) sin( x) 1 Number of solutions: b) 1 sin( x) Number of solutions: 4 c) sin( x) 0 Number of solutions: d) sin( x) 5 Number of solutions: 8

29 Example: How many solutions are there for the following equations OVER ONE PERIOD? a) cos( x) 1 Number of solutions: b) 4cos( x) 1 Number of solutions: c) 4cos( x) Number of solutions: 9

30 Example: How many solutions are there for the following equation on the number line? a) 1 cos( x) Number of solutions: b) cos( x) 4 Number of solutions: 30

31 Note: An equation of the form sin( x) period if 1a 1. An equation of the form sin( x) period if a 1 or a 1 a or cos( x) a or cos( x) a has TWO solutions over one a has NO solutions over one For the cases where a 0, 1 or -1, think about the graph to answer. Pay attention to the end points of the interval. If the interval given on the problem covers more than one period, adjust your answer accordingly. 31

32 POPPER for Section 6.3: Question#4: How many solutions are there for the equation 4cos( ) 5 6 0,. x over the interval a) 0 b) 1 c) d) 3 e) Infinitely many solutions f) None of these 3