MATHEMATICIAN DATE BAND

BASIC TRIGONOMETRIC EQUATIONSPRECALCULUS | PACKER COLLEGIATE INSTITUTE

We’re going to look at solving basic trigonometric equations from three viewpoints: the unit circle, the graph, and the calculator. Use the following images to help you: the unit circle, and the graphs of sine and cosine. You will need to use a protractor for the unit circle.

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Sine and Cosine graphs in degrees

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Part I: Unit Circles, Graphs, and Calculators, Oh my!

For all of these problems, we’re going to be looking for answers from .

1. Let’s solve in multiple ways.

(a) The two solutions are angles in Quadrants I and II. We know this because:

(b) On this mini-unit circle, mark approximately where the two solutions are:

(c) Use your protractor and the large unit circle to find the first quadrant solution, approximately:

(d) Use the first quadrant answer and logic to find the second quadrant solution, approximately: Do not measure this answer using a protractor!

Work:

(e) Find the two solutions using the sine graphs provided… and

Are they close to your answers in parts (c) and (d)? YES / NO

(f) Now let’s use your calculator. Make sure it is in degree mode… Now check your approximate answers found above by typing “sin([your approximate answers here])”. Do you get an output of about 0.3? YES / NO

(g) You saw this in Geometry, but it was a long time ago! We can use your calculator to find a better approximate

answer to . To “undo” the sine function, we took the inverse sine of both sides. So we typed:

What is the output (write five decimal places)?

Notice that this output is in the first quadrant.

Do the same calculation you did in part (d) to get the second quadrant answer (write five decimal places):

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2. Let’s solve in multiple ways.

(a) The two solutions are angles in Quadrants III and IV. We know this because:

(b) On this mini-unit circle, mark approximately where the two solutions are:

(c) Use your protractor and the large unit circle to find the third quadrant solution, approximately:

(d) Use the first quadrant answer and logic to find the fourth quadrant solution, approximately: Do not measure this answer using a protractor!

Work:

(e) Find the two solutions using the sine graphs provided… and

Are they close to your answers in parts (c) and (d)? YES / NO

(f) Now let’s use your calculator. Make sure it is in degree mode… Now check your approximate answers found above by typing “sin([your approximate answers here])”. Do you get an output of about -0.3? YES / NO

(g) Using your calculator, find one answer. What is the output (write five decimal places)?

Notice that this output is in the fourth quadrant.

Do the same calculation you did in part (d) to get the third quadrant answer (write five decimal places):

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3. Let’s solve in multiple ways.

(a) The two solutions are angles in Quadrants I and IV. We know this because:

(b) On this mini-unit circle, mark approximately where the two solutions are:

(c) Use your protractor and the large unit circle to find the first quadrant solution, approximately:

(d) Use the third quadrant answer and logic to find the fourth quadrant solution, approximately: Do not measure this answer using a protractor!

Work:

(e) Find the two solutions using the cosine graphs provided… and

Are they close to your answers in parts (c) and (d)? YES / NO

(f) Now let’s use your calculator. Make sure it is in degree mode… Now check your approximate answers found above by typing “cos([your approximate answers here])”. Do you get an output of about 0.8? YES / NO

(g) Using your calculator, find one answer. What is the output (write five decimal places)?

Notice that this output is in the first quadrant.

Use similar logic as in part (d) to get the fourth quadrant answer (write five decimal places):

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4. Let’s solve in multiple ways.

(a) The two solutions are angles in Quadrants II and III. We know this because:

(b) On this mini-unit circle, mark approximately where the two solutions are:

(c) Use your protractor and the large unit circle to find the second quadrant solution, approximately:

(d) Use the first quadrant answer and logic to find the third quadrant solution, approximately: Do not measure this answer using a protractor!

Work:

(e) Find the two solutions using the cosine graphs provided… and

Are they close to your answers in parts (c) and (d)? YES / NO

(f) Now let’s use your calculator. Make sure it is in degree mode… Now check your approximate answers found above by typing “cos([your approximate answers here])”. Do you get an output of about -0.8? YES / NO

(g) Using your calculator, find one answer. What is the output (write five decimal places)?

Notice that this output is in the second quadrant.

Use similar logic as in part (d) to get the third quadrant answer (write five decimal places):

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5. Let’s solve in multiple ways.

(a) The two solutions are angles in Quadrants I and III. We know this because:

(b) On this mini-unit circle, mark approximately where the two solutions are:

(c) Use your protractor and the large unit circle to find the first quadrant solution, approximately:

(d) Use the first quadrant answer and logic to find the third quadrant solution, approximately: Do not measure this answer using a protractor!

Work:

(e) Now let’s use your calculator. Make sure it is in degree mode… Now check your approximate answers found above by typing “tan([your approximate answers here])”. Do you get an output of about 1.1? YES / NO

(f) Using your calculator, find one answer. What is the output (write five decimal places)?

Notice that this output is in the first quadrant.

Use similar logic as in part (d) to get the third quadrant answer (write five decimal places):

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6. Let’s solve in multiple ways.

(a) The two solutions are angles in Quadrants II and IV. We know this because:

(b) On this mini-unit circle, mark approximately where the two solutions are:

(c) Use your protractor and the large unit circle to find the second quadrant solution, approximately:

(d) Use the first quadrant answer and logic to find the fourth quadrant solution, approximately: Do not measure this answer using a protractor!

Work:

(e) Now let’s use your calculator. Make sure it is in degree mode… Now check your approximate answers found above by typing “tan([your approximate answers here])”. Do you get an output of about -1.1? YES / NO

(f) Using your calculator, find one answer. What is the output (write five decimal places)?

Notice that this output is in the fourth quadrant.

Use similar logic as in part (d) to get the second quadrant answer (write five decimal places):

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Part II: Focusing on your calculator!

For all of these problems, we’re going to be looking for answers from . Rely on your calculator!

7. Mark where the solutions are for

approximately on this mini-unit circle.

Use your calculator to find one solution. Round it to five decimal places.

Now use that answer and logic to find the second solution (to five decimal places). Show your work.

8. Mark where the solutions are for

approximately on this mini-unit circle.

Use your calculator to find one solution. Round it to five decimal places.

Now use that answer and logic to find the second solution (to five decimal places). Show your work.

9. Mark where the solutions are for

approximately on this mini-unit circle.

Use your calculator to find one solution. Round it to five decimal places.

Now use that answer and logic to find the second solution (to five decimal places). Show your work.

10. Mark where the solutions are for

approximately on this mini-unit circle.

Use your calculator to find one solution. Round it to five decimal places.

Now use that answer and logic to find the second solution (to five decimal places). Show your work.

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11. Mark where the solutions are for

approximately on this mini-unit circle.

Use your calculator to find one solution. Round it to five decimal places.

Now use that answer and logic to find the second solution (to five decimal places). Show your work.

12. Mark where the solutions are for

approximately on this mini-unit circle.

Use your calculator to find one solution. Round it to five decimal places.

Now use that answer and logic to find the second solution (to five decimal places). Show your work.

Part III: Solving More Complex Equations Using Your Calculator

For all of these problems, we’re going to be initially looking for answers from . I will provide a mini-unit circle to help you in each problem. Write your answers rounded to the second decimal place.

13. Work:

Solutions for : and

Okay, we originally had our solutions restricted from . But now I want all possible solutions:

All possible solutions: and

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14. Work:

Solutions for : and

All possible solutions: and

15. Work:

Solutions for : and

All possible solutions: (Hint: notice for this tangent equation I am only leaving one blank here! You can consolidate all solutions easily into one expression! Think about why!)

16. Work:

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Solutions for : and

All possible solutions: and

17. Work:

Solutions for : and

All possible solutions: and

18. Work:

Solutions for : and

All possible solutions: (Hint: see above for #15!)

19. Work:

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Solutions for : and

All possible solutions: and

Part IV: Solving More Complex Equations Using Desmos!

20. Okay we’re going to use Desmos to solve #13-#19! Make sure Desmos is in degree mode!

#13:

Graph on

Desmos.

Find an appropriate window (fill in the y range as well as the step value)

Look for the solutions for

. Write them down, rounded to two decimal points:

Do they match your work from

#13? YES / NO

#14:

Graph the appropriate

equations on Desmos.

Find an appropriate window (fill in the y range as well as the step value)

Look for the solutions for

. Write them down, rounded to two decimal points:

Do they match your work from

#14? YES / NO

#15:

Graph the appropriate

equations on Desmos.

Find an appropriate window (fill in the y range as well as the step value)

Look for the solutions for

. Write them down, rounded to two decimal points:

Do they match your work from

#15? YES / NO

#16:

Graph the appropriate

equations on Desmos.

Find an appropriate window (fill in the y range as well as the step value)

Look for the solutions for

. Write them down, rounded to two decimal points:

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Do they match your work from

#16? YES / NO

#17:

Graph the appropriate

equations on Desmos.

Find an appropriate window (fill in the y range as well as the step value)

Look for the solutions for

. Write them down, rounded to two decimal points:

Do they match your work from

#17? YES / NO

#18:

Graph the appropriate

equations on Desmos.

Find an appropriate window (fill in the y range as well as the step value)

Look for the solutions for

. Write them down, rounded to two decimal points:

Do they match your work from

#18? YES / NO

#19:

Graph the appropriate

equations on Desmos.

Find an appropriate window (fill in the y range as well as the step value)

Look for the solutions for

. Write them down, rounded to two decimal points:

Do they match your work from

#19? YES / NO

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