Trigonometry : Find all angles in a range given specific output

Study concepts, example questions & explanations for Trigonometry

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Example Questions

Example Question #1 : Find All Angles In A Range Given Specific Output

Find all angles  between and  when .

Possible Answers:

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

This problem relies on understanding reference angles and coterminal angles. A reference angle  for an angle  in standard position is the positive acute angle between the x axis and the terminal side of the angle . A table of reference angles for each quadrant is given below.

Screen shot 2020 07 30 at 11.05.57 am

Since  is negative, solutions for  will be in Quadrants II and III because these are the quadrants where cosine is negative.

Use inverse cosine and a calculator to find :

In Quadrant II, we have , so .

In Quadrant III, , so 

Therefore  and .

Example Question #2 : Find All Angles In A Range Given Specific Output

Find all angles  between and  when .

Possible Answers:

 and 

and 

 and 

Correct answer:

and 

Explanation:

This problem relies on understanding reference angles and coterminal angles. A reference angle  for an angle  in standard position is the positive acute angle between the x axis and the terminal side of the angle . A table of reference angles for each quadrant is given below.

Screen shot 2020 07 30 at 11.05.57 am

Since  is negative, solutions for  will be in Quadrants II and IV because these are the quadrants where tangent is negative. Use inverse tangent and a calculator to find :

In Quadrant II, we have ,  so .

In Quadrant IV, , so .

Therefore and .

Example Question #3 : Find All Angles In A Range Given Specific Output

Find all angles  when .

Possible Answers:

 and 

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

We can use reference angles, inverse trig, and a calculator to solve this problem. Below is a table of reference angles. 

Screen shot 2020 07 30 at 11.05.57 am

We have  so . Next, think about where sine is negative, or reference the Function Signs column of the above table. Sine is negative in Quadrants III and IV.

In Quadrant III, .

In Quadrant IV, .

If this problem asked for values of  between  and , our work would be done, but this problem does not restrict the range, so we need to give all possible values of  by generalizing our answers. To do this, we must understand that all angles that are coterminal to  and  will also be solutions. Coterminal angles add or subtract multiples of . To write this generally, we write:

 and .

Example Question #4 : Find All Angles In A Range Given Specific Output

Find all angles  when .

Possible Answers:

 and 

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

We can use reference angles, inverse trig, and a calculator to solve this problem. Below is a table of reference angles. 

Screen shot 2020 07 30 at 11.05.57 am

We have , so  . Next, think about where tangent is positive, or reference the Function Signs column of the above table. Tangent is positive in Quadrants I and III.

In Quadrant I, .

In Quadrant III, .

If this problem asked for values of  between  and , our work would be done, but this problem does not restrict the range, so we need to give all possible values of  by generalizing our answers. To do this, we must understand that all angles that are coterminal to  and  will also be solutions. Coterminal angles add or subtract multiples of . To write this generally, we write:

 and 

Example Question #5 : Find All Angles In A Range Given Specific Output

Find all positive values of  less than  for which .

Possible Answers:

 and 

 and 

Correct answer:

 and 

Explanation:

At first glance, you may think that this problem has infinite answers, since there would be infinitely many negative coterminal angles that could satisfy this; however, notice that the question asks only for positive values of . In other words, this question is simply asking for values of  between  and  that satisfy this equation.

First, let's think about where the cosine function is negative. Per the chart below, it will be in Quadrants II and III.

Screen shot 2020 07 30 at 10.25.02 am

The reference angle for each angle solution will have its cosine equal to  and is . Consult the chart of reference angles below for Quadrants II and III:

Screen shot 2020 07 30 at 11.05.57 am

QII: 

QIII: 

Example Question #6 : Find All Angles In A Range Given Specific Output

Find all angles  between and  when .

Possible Answers:

 and 

  and 

Correct answer:

  and 

Explanation:

This problem relies on understanding reference angles and coterminal angles. A reference angle  for an angle  in standard position is the positive acute angle between the x axis and the terminal side of the angle . A table of reference angles for each quadrant is given below.

Screen shot 2020 07 30 at 11.05.57 am

Since  is positive, solutions for  will be in Quadrants I and IV because these are the quadrants where cosine is positive. Use inverse cosine and a calculator to find :

In Quadrant I, we have , so .

In Quadrant IV, , so .

Therefore  and .

Example Question #7 : Find All Angles In A Range Given Specific Output

Find all angles  between and  when .

Possible Answers:

  and 

   and 

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

This problem relies on understanding reference angles and coterminal angles. A reference angle  for an angle  in standard position is the positive acute angle between the x axis and the terminal side of the angle . A table of reference angles for each quadrant is given below.

Screen shot 2020 07 30 at 11.05.57 am

Since  is positive, solutions for  will be in Quadrants I and II because these are the quadrants where sine is positive. Use inverse sine and a calculator to find :

In Quadrant I, we have , so .

In Quadrant II, , so 

Therefore  and .

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