Pre-Class Problems20 for Monday, April 2

These are the type of problems that you will be working on in class. These problems are fromLesson 10.

Solution to Problems on the Pre-Exam.

You can go to the solution for each problem by clicking on the problemletter.

Objective of the following problems: To find all the solutions to a trigonometric equation in a specified interval.

1a.Find all the exact solutions (in radians) of the equation in the interval. Put a box around your answer(s).

1b.Find all the exact solutions (in degrees) of the equation in the interval.

1c.Find all the exact solutions (in radians) of the equation in the interval.

1d.Find all the exact solutions (in degrees) of the equation

in the interval.

1e.Find all the exact solutions (in radians) of the equation in the interval.

1f.Find all the exact solutions (in degrees) of the equation in the interval.

1g.Find all the exact solutions (in radians) of the equation in the interval.

Additional problems available in the textbook: Page 618 …19–32. Examples 2, 3, and 4starting on page 609.

Solutions:

1a. , Back to Problem 1.

Let . Then

NOTE: Cosine is negative in the second and third quadrants.

To find the reference angle :

NOTE: The one solution in the second quadrant that is in the interval is given by .

All solutions for u in II: , where n is an integer

All solutions for x in II: and

, where n is an integer.

Simplifying, we have that = = = , where n is an integer

The solutions for which are in the interval :

:

:

:

NOTE: . So, this solution is not in the interval .

:

:

NOTE: . So, this solution is not in the interval .

NOTE: The one solution in the third quadrant that is in the interval is given by .

All solutions for u in III: , where n is an integer

All solutions for x in III: and

, where n is an integer.

Simplifying, we have that = = = , where n is an integer

The solutions for which are in the interval :

:

:

NOTE: . So, this solution is not in the interval .

:

:

:

NOTE: . So, this solution is not in the interval .

Answer:

1b. , Back to Problem 1.

Let . Then

NOTE: Sine is positive in the first and second quadrants.

To find the reference angle :

NOTE: The one solution in the first quadrant that is in the interval

is given by .

All solutions for u in I: , where n is an integer

All solutions for in I: and

, where n is an integer.

The solutions for which are in the interval

:

:

:

:

:

NOTE: . So, this solution is not in the interval

.

:

:

NOTE: . So, this solution is not in the interval .

NOTE: The one solution in the second quadrant that is in the interval

is given by .

All solutions for u in II: , where n is an integer

All solutions for in II: and

, where n is an integer.

The solutions for which are in the interval

:

:

:

:

NOTE: . So, this solution is not in the interval

.

:

:

NOTE: . So, this solution is not in the interval .

Answer:

1c. , Back to Problem 1.

Let . Then

NOTE: Tangent is negative in the second and fourth quadrants.

To find the reference angle :

NOTE: The one solution in the second quadrant that is in the interval is given by .

All solutions for u in II: , where n is an integer

All solutions for in II: and

,

where n is an integer.

Simplifying, we have that = = = , where n is an integer

The solutions for which are in the interval :

:

NOTE: . So, this solution is not in the interval .

:

:

:

NOTE: . So, this solution is not in the interval .

:

NOTE: . So, this solution is not in the interval .

NOTE: The one solution in the fourth quadrant that is in the interval is given by .

All solutions for u in IV: , where n is an integer

All solutions for in IV: and

,

where n is an integer.

Simplifying, we have that = =

= , where n is an integer

The solutions for which are in the interval :

:

NOTE: . So, this solution is not in the interval .

:

:

:

NOTE: . So, this solution is not in the interval .

:

NOTE: . So, this solution is not in the interval .

Answer:

1d. , Back to Problem 1.

Let . Then

NOTE: Cosine is positive in the first and fourth quadrants.

To find the reference angle :

NOTE: The one solution in the first quadrant that is in the interval

is given by .

All solutions for u in I: , where n is an integer

All solutions for in I: and

, where n is an integer.

The solutions for which are in the interval

:

:

:

:

:

NOTE: . So, this solution is not in the interval

.

:

:

:

:

:

:

:

NOTE: . So, this solution is not in the interval .

NOTE: The one solution in the fourth quadrant that is in the interval

is given by .

All solutions for u in IV: , where n is an integer

All solutions for in IV: and

, where n is an integer.

The solutions for which are in the interval

:

:

:

:

:

NOTE: . So, this solution is not in the interval

.

:

:

:

:

:

:

:

NOTE: . So, this solution is not in the interval .

Answer:

1e. , Back to Problem 1.

Let . Then

NOTE: Sine is negative in the third and fourth quadrants.

To find the reference angle :

NOTE: The one solution in the third quadrant that is in the interval is given by .

All solutions for u in III: , where n is an integer

All solutions for x in III: and

, where n is an integer.

NOTE:

Simplifying, we have that = =

= , where n is an integer

The solutions for which are in the interval :

:

:

:

:

NOTE: . So, this solution is not in the interval .

:

:

NOTE: . So, this solution is not in the interval .

NOTE: The one solution in the fourth quadrant that is in the interval is given by .

All solutions for u in IV: , where n is an integer

All solutions for x in IV: and

, where n is an integer.

NOTE:

Simplifying, we have that = =

= , where n is an integer

The solutions for which are in the interval :

:

:

:

:

:

NOTE: . So, this solution is not in the interval .

:

:

NOTE: . So, this solution is not in the interval .

Answer:

1f. , Back to Problem 1.

Let . Then

NOTE: The angle solutions for this equation must lie on one of the coordinate axes since zero is not positive nor negative.

NOTE: Using Unit Circle Trigonometry, we know that . Thus, the one solution, which lies on the positive x-axis, is .

All solutions for u on the positive x-axis: , where n is an integer

NOTE: Using Unit Circle Trigonometry, we know that . Thus, the one solution, which lies on the negative x-axis, is .

All solutions for u on the negative x-axis: , where n is an integer

NOTE: These two solutions of and , where n is an integer, may be written as , where n is an integer.

All solutions for : and

, where n is an integer.

The solutions for which are in the interval :

:

:

:

:

:

:

:

:

:

:

Answer:

1g. , Back to Problem 1.

Let . Then

NOTE: The angle solutions for this equation must lie on one of the coordinate axes since is the minimum negative number in the range of the cosine function.

NOTE: Using Unit Circle Trigonometry, we know that . Thus, the one solution, which lies on the negative x-axis, is .

All solutions for u on the negative x-axis: , where n is an integer

All solutions for : and

,

where n is an integer.

Simplifying, we have that = =

= , where n is an integer

The solutions for which are in the interval :

:

:

:

:

:

:

NOTE: . So, this solution is not in the interval .

:

:

:

:

NOTE: . So, this solution is not in the interval .

Answer:

Solution to Problems on the Pre-Exam:Back to Page 1.

25.Find all the solutions (in degrees) to the equation in the interval . (9 pts.) Put a box around your answer(s).

Let . Then

NOTE: Cosine is negative in the second and third quadrants.

To find the reference angle :

NOTE: The one solution in the second quadrant that is in the interval is given by .

All solutions for u in II: , where n is an integer

All solutions for x in II: and

, where n is an integer.

The solutions for which are in the interval :

:

:

:

:

:

NOTE: The one solution in the third quadrant that is in the interval is given by .

All solutions for u in III: , where n is an integer

All solutions for x in III: and

, where n is an integer.

:

:

:

:

: