trigonometric equations solving for the angle (the second of two note days and a work day) (6.2)(2)

Trigonometric Equations

Solving for the angle

(The second of two note days and a work day)

(6.2)(2)

POD

True/ false:

The x-intercepts of y = sin 2x are ±πn/2.

POD

True/ false:

The x-intercepts of y = sin 2x are π/2 ±πn/2.

There are couple of ways to do this.

1. Set sin 2x = 0, and solve.

2. Consider the x-intercepts of the graph of y = sin x. We’ve changed something in the equation—how does that change the graph? What x-scale could help see it?

Pick up from last time

Solve by factoring.

What are the four steps again?

0tantansin4 2 xxx

Pick up from last time

Step one: Isolate trig functions by factoring.

0)1sin2)(1sin2(tan

0)1sin4(tan

0tantansin42

2

xxx

xx

xxx

Pick up from last time

Start step 2: use inverse trig

functions.

0)1sin2)(1sin2(tan

0)1sin4(tan

0tantansin42

2

xx

xx

xxx

0

0tan

x

x

6

2

1sin

1sin2

01sin2

x

x

x

x

62

1sin

2

1sin

1sin2

01sin2

1

x

x

x

x

Pick up from last time

Step two: Find anglesin one rotation.

Note: the inverse trig function for the last one gives us a value outside the interval 0 ≤ θ ≤ 2π. If we use that tool, we have to build on it.

x

x 0tan

6

5,

6

2

1sin

x

x

6

11,

6

7

62

1sin

2

1sin

1

x

x

x

Pick up from last time

Step three: Find the general solution (all angles).

We can combine those last four into this.

Or this.

nnx

6

5,

6

nx nnx 2

6

5,2

6 nnx

26

11,2

6

7

nx

6

nx

nx

Solving for the variable

Solve for u. (Note, this is not 2 csc4 u.) You could replace the 2u with x or θ for now, if it helps.

042csc4 u

Solving for the variable

Step one: Isolate trig functions by factoring.

The second factor does not provide a meaningful solution.

0)22)(csc22(csc

042csc22

4

uu

u

2

2

2

12sin

22sin

1

22csc

22csc2

u

u

u

u

22csc

22csc2

u

u

Solving for the variable

Step two: Find angles in one rotation. Again, we build off of the angle we get for the negative sine value.

2

22sin u

4

3,

42

2

2sin2 1

u

u

4

7,

4

52

2

2sin2 1

u

u

Solving for the variable

Step three: Find all angles.

Combined, what would the statement be?

2

22sin u

nnu

u

24

3,2

42

4

3,

42

nnu

u

24

7,2

4

52

4

7,

4

52

Solving for the variable

Step three: Find all angles.

Combined, what would the statement be?

We can use this to solve for u. We can also use the separate statements to solve for u. We’ll get the same answer.

nu24

2

Solving for the variable

Step four: Solve for u with all statements.

This is a mouthful. What’s the easier way to present it?

nnu

u

nnu

u

8

3,

8

8

3,

8

24

3,2

42

4

3,

42

nnu

u

nnu

u

8

7,

8

58

7,

8

5

24

7,2

4

52

4

7,

4

52

Solving for the variable

An elegant combination:

We could get this using that single statement from before.

nnu

u

24

3,2

42

4

3,

42

nnu

u

24

7,2

4

52

4

7,

4

52

nu

nu

48

242

Approximating solutions

Group and factor this to start.

Work in degrees for a switch.

06sin3tan10tansin5

Approximating solutions

Step one: Group and factor this to start.

0)2)(sin3tan5(

0)2(sin3)2(sintan5

0)6sin3()tan10tansin5(

06sin3tan10tansin5

Approximating solutions

Step two: Find angles in one rotation.

Once again, we have a factor that does not provide a meaningful solution.

0)2)(sin3tan5(

329180149

14918031

315

3tan

5

3tan

3tan5

1

2sin

Approximating solutions

Step three: Find all angles.

0)2)(sin3tan5(

n180149

Using a graph

Graph to find the roots of this equation in one rotation.

What patterns do you see?

What x-scale could help to see the intercepts?

03sin2sinsin xxx

Using a graph

Graph to find the roots of this equation in one rotation.

Now graph it -2π to 2π. What do you see?

2,

2

3,

3

4,,

3

2,

2,0

28.6,71.4,19.4,14.3,09.2,57.1,0

03sin2sinsin

x

x

xxx

Using a graph

What about this one? How does it compare to the one before?

13sin2sinsin xxx

Using a graph

What about this one? x = .171, 1.24.What do you see here?

If we rewrote the equation to set one side to 0, what would we have done to the graph?

13sin2sinsin xxx

Application

From example 9, page 476.

What do the various elements of this equation signify?

12)79(365

2sin3)(

ttD

Application

From example 9, page 466.

How many days have more than 10.5 hours?

What’s a fast way to find out?

12)79(365

2sin3)(

ttD

Application

From example 9, page 466.

How many days have more than 10.5 hours?

Graph it first on calculators and calculate intersections.

12)79(365

2sin3)(

ttD

Application

From example 9, page 466.

How many days have more than 10.5 hours?

Graph it first on calculators and calculate intersections.

x = 48.6 and 291.9

Now, algebraically.

12)79(365

2sin3)(

ttD

Application

How many days have more than 10.5 hours?

The expression is actually the angle.

5.)79(365

2sin

5.1)79(365

2sin3

5.1012)79(365

2sin3

t

t

t

)79(365

2t

Application

The expression is actually the angle.

Substitute θ for this expression, and solve for θ.

sinθ = -.5

θ = -π/6 ≈ -.52 θ = π/6 + π ≈ 3.67

Un-substitute and solve for t.

)79(365

2t

Application

Un-substitute and solve for t.

Find the difference between the days to answer the question (finally). Graphing was easier…

8.48

21.302

36552.79

52.)79(365

2

t

t

t

2.292

2.2132

36567.379

67.3)79(365

2

t

t

t