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Trigonometry Unit

Lesson Plans > Mathematics > Algebra > Functions > Trigonometry

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A car travels 20 miles south, and then 40 miles east. How far, and in what direction, does the car end up from where it started?

I hiked to the top of a conical mountain which had an elevation of 4000 feet. If the angle of inclination of the mountain is 10 degrees, and I hiked in a straight line, how far did I hike?

A pilot is in an airplane that is flying at an elevation of 2,000 feet. His angle of declination to the runway is 5 degrees. What is the horizontal distance from the plane to the runway?

A flagpole casts a shadow which is half the length of the flagpole. What is the sun's angle of inclination?

A car travels due north, and then travels due west. The car ends up 30 miles from its starting point, and 40 degrees North of West. How many miles west of its starting point did the car end up?

A conical mountain's base is five times as long as its altitude. What is the mountain's angle of elevation?

A building is 50 feet wide, 50 feet long, and 80 feet tall. The sun is to the east of the building, and the distance from the tip of the building's shadow to the eastern face of the building is 120 feet. What is the sun's angle of inclination?

Two flagpoles differ in height by 5 feet. The lengths of their shadows are 20 feet and 24 feet. What is the sun's angle of inclination?

Two cars start at the same point and drive east. The first car drives 10 miles further than the other. Then both cars turn south, and the first car drives 8 miles further than the second car. Both cars end up in the same direction from their starting point. What is their direction from their starting point?

Mr. T hiked in a straight line 6.25 miles up the east side of a mountain, and 13 miles down the west side. He ended up 15.75 miles from his starting point. What is the difference between the angle of inclination on the east side and the angle of inclination on the west side?

Trigonometry Unit

This trigonometry unit is not for everyone. I specifically designed it for students who are currently taking algebra two, and are planning to take physics during the coming year. The pre-cal course I teach covers trigonometry in great depth, but that doesn't help students who will be taking pre-cal and physics concurrently. this unit focuses entirely on the more "practical" elements of trigonometry that they'll use in physics.

We start with a basic refresher on triangles (because some of my algebra two students have not yet taken geometry!). Then we do basic trig function calculations based on various right triangles (mostly Pythagorean Triple triangles). Then we look at inverse trig functions, and use everything we've learned to solve triangles. Finally, putting it all together, we solve word problems involving trigonometry.

The second word problem worksheet involves compass directions. This gives students practice in writing a displacement (a distance and direction), which is valuable for physics students. Note that the answer key gives answer in the "degrees north of east" notation; if you want to use a different notation, you'll want to modify the answer key to suit your needs.

Finally, there is a handout that gives the problems in the slides. You can print these and hand them to your students so they don't have to copy the word problems into their notes.

Lesson by Mr. Twitchell

Triangles and Labeling

1. Draw a triangle and label it ABC. Which sides are adjacent to angle A? Which side is opposite side A?

2. Draw a triangle and label it WXY. Which sides are adjacent to angle W? Which side is opposite side W?

3. Draw a right triangle and label it MNQ, with the right angle and Q. Which side is the hypotenuse? Which leg is adjacent to M? Which leg is opposite M?

4. Draw a right triangle and label it ABC, with the right angle at A. Which side is the hypotenuse? Which leg is adjacent to C? Which leg is opposite C?

5. Draw and label a right triangle with side lengths f = 5, g = 12, and k = 13. Which angle is the right angle? The shortest leg is adjacent to which angle?

6. Draw and label a right triangle with side lengths r = 3, s = 5, t = 4. Which side is the hypotenuse? Which angle is the right angle? The longest leg is opposite which angle?

7. Draw and label right triangle ABC, with right angle is at C, and A = 30º. What is the measure of angle B? Which leg is opposite angle A?

8. In triangle XYZ, X = 35º and Y = 55º. Is this a right triangle? Draw and label the triangle.

9. In right triangle MAP, the legs are m = 6 and p = 8. Find the length of the hypotenuse. Draw and label the triangle. Which leg is adjacent to M?

10. In right triangle COW, with right angle at W, w = 26 and c = 10. Find the length of o. Draw and label the triangle. Which side is the hypotenuse? Which leg is adjacent to C?

11. The sides of a triangle are a = 7, b = 9, and c = 11. Is this a right triangle? Draw and label the triangle. If it is a right triangle, which side is the hypotenuse?

12. The sides of a triangle are a = 20, b= 16, and c = 12. Is this a right triangle? Draw and label the triangle. If it is a right triangle, which side is the hypotenuse?

13. Two of the angles of triangle FGH are F = 40º and G = 50º. Is this a right triangle? Draw and label the triangle. if it is a right triangle, which side is the hypotenuse?

14. Triangle ABC has right angle at C, and A = 20º. Draw and label the triangle. Find the value of B. Which leg is adjacent to B? Which side is the hypotenuse?

15. In triangle JAC, J = 45º, A = 45º, and j = 10. Find the values of C, a, and c. Draw and label the triangle.

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Finding and Using Trig Values

1. In right triangle ABC, the lengths of the legs are a = 3 and b = 4. Find sin A, cos A, and tan A.

2. In right triangle BCD, with right angle at B, c = 5 and d = 12. Find sin C and cos C.

3. In right triangle LMN, the length of hypotenuse is m = 10, and n = 7.071. Find tan L, sin L, and cos L.

4. The sides of right triangle XYZ are x = k, y = k + 3, and z = k + 6. Find sin X, cos X, and tan X.

5. The two legs of right triangle ABC are a = k and b = 3k. Find tan A and tan B.

6. The shorter leg of right triangle DEF is d = 7, and the hypotenuse length is one more than the longer leg. Find tan D, sin D, and cos D.

7. The hypotenuse of right triangle CDE is one more than one of its legs, and 8 more than the other leg. What is the tangent of the angle opposite the shorter leg?

8. In right triangle FED, with right angle at E, sin F = sin D, and e = 5. Find the lengths of f and d.

9. In right triangle ABC, with right angle at C, sin A =
3
5
, and a = 12. Find c.

10. In right triangle KHM, with right angle at M, tan K =
9
40
, and h = 120. Find the length of the hypotenuse.

11. In right triangle JKL, with right angle at J, tan K =
12
5
, and the length of the hypotenuse is 52. What are the lengths of the legs?

12. In right triangle ABC, with right angle at C, cos A =
3
5
. Find cos B.

13. In right triangle MNO, with right angle at M, cos N =
4
7
. Find cos O.

14. In right triangle TUV, the legs are t = 14 and u = 48. Find sin T, cos T, and tan T.

15. In right triangle MAT, one leg is m = 10, and the hypotenuse is T = 26. Find sin A, cos A, and tan A.

Finding and Using Trig Values: Answer Key

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Solving Right Triangles

For each triangle, list all the missing side lengths and angle measures.

1. ABC: C = 90º, A = 30º, a = 20

2. XYZ: X = 90º, Y = 20º, x = 10

3. MNO: M = 90º, N = 40º, m = 15

4. ABC: A = 30º, B = 60º, a = 5

5. RTY: R = 90º, T = 5º, t = 10

6. SAM: S = 90º, A  = 60º, m = 100

7. WAT: W = 90º, A  = 50º, w = 50

8. BAT: B = 90º, T  = 72º, t = 10

9. ABC: A  = 15º; B  = 75º, c = 12

10. ABC: C  = 90º, a= 5, b = 6

11. XMQ: X  = 90º; x = 20, m = 15

12. VBX: V = 90º, b = 10, x = 10

13. FED: D = 90º, d = 50, e = 30

14. POW: P = 90º, p = 20, w = 10

15. ABC: C = 90º, c = a + 2, and b = a + 1

16. XYZ:  X = 90º, the hypotenuse has length 20, and y is twice as long as z.

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Trigonometry Word Problems

Draw a diagram, label it with known information and variables representing unknowns, and the solve the problem.

1. A car drives 50 miles north, and then 120 miles east. How far from the car’s starting point did the car end up?

2. A car drives 30 miles east, and a few miles south, and ends up 40 degrees south of east from his starting point. How far south did the car travel?

3. After traveling 40 miles, partly in a westerly direction, and partly in a northerly direction, a car is 45 degrees north of west from its original starting point. What is the car's distance from its starting point?

4. A flag pole is 20 feet tall, and the sun’s angle of inclination is 40 degrees. How long is the flag’s shadow?

5. The angle of inclination of a conical mountain is 10 degrees, and the mountain’s height is 4000 feet. How far would you have to walk to reach the summit?

6. A car ended up 500 miles away from its starting point, after traveling 480 miles north, and then x miles west. How far did the car travel west?

7. A man travels 5 km at 30 degrees north of east. How far north of his starting point does he end up?

8. A cone’s slant height is 20 inches. The cone’s base angle is 80 degrees. How tall is the cone?

9. A cone’s base radius is 8 inches, and its height is 6 inches. What is its slant height?

10. A pole is 12 feet tall, and its shadow is 3 feet long. What is the distance between the top of the pole and the far end of its shadow?

11. What is the sun's angle of inclination, if the pole is 30 feet tall, and it casts a 12 foot shadow?

12. A cone's diameter is equal to its height. What is base angle of the cone?

13. A roof is sloped such that there is a 12 inch horizontal run for each 8 inch vertical rise. What is the angle of inclination of the roof?

14. A roof has a slope of 30 degrees on the west side of the building, and a slope of 45 degrees on the east side of the building. If the building is 40 feet wide from east to west, how many feet off-center will the peak of the roof be?

15. A child slides down a 20 foot slide with a 35 degree angle of inclination.  How much distance does the child travel getting from the bottom of the slide back to the top (assuming he walks to the bottom of the vertical ladder and then climbs it)?

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Compass Direction Problems

1. A car travels 3 miles east and 4 miles north. What is the car's displacement?

2. A car travels 5 miles north, and then a few miles west. It ends up 13 miles from its starting point. What is the car's displacement?

3. A car travels 5 miles west and a few miles south. It ends up 10 miles from its starting point. What is the car's displacement?

4. A car travels x miles east, and the same number of miles south. What is the car's displacement?

5. A car travels x miles east and twice as far north. What is the car's displacement?

6. A car travels 10 miles east, 5 miles south, and 15 miles west. What is the car's displacement?

7. A car travels 20 miles east, 10 miles north, 16 miles west, and 7 miles south. What is the car's displacement?

8. A car travels the same distance west as it travels south, and ends up 20 miles from its starting point. What is the car's displacement?

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Problems Handout

1. A car travels 20 miles south, and then 40 miles east. How far, and in what direction, does the car end up from where it started?

2. I hiked to the top of a conical mountain which had an elevation of 4000 feet. If the angle of inclination of the mountain is 10 degrees, and I hiked in a straight line, how far did I hike?

3. A pilot is in an airplane that is flying at an elevation of 2,000 feet. His angle of declination to the runway is 5 degrees. What is the horizontal distance from the plane to the runway?

4. A flagpole casts a shadow which is half the length of the flagpole. What is the sun's angle of inclination?

5. A car travels due north, and then travels due west. The car ends up 30 miles from its starting point, and 40 degrees North of West. How many miles west of its starting point did the car end up?

6. A conical mountain's base is five times as long as its altitude. What is the mountain's angle of elevation?

7. A building is 50 feet wide, 50 feet long, and 80 feet tall. The sun is to the east of the building, and the distance from the tip of the building's shadow to the eastern face of the building is 120 feet. What is the sun's angle of inclination?

8. Two flagpoles differ in height by 5 feet. The lengths of their shadows are 20 feet and 24 feet. What is the sun's angle of inclination?

9. Two cars start at the same point and drive east. The first car drives 10 miles further than the other. Then both cars turn south, and the first car drives 8 miles further than the second car. Both cars end up in the same direction from their starting point. What is their direction from their starting point?

10. Mr. T hiked in a straight line 6.25 miles up the east side of a mountain, and 13 miles down the west side. He ended up 15.75 miles from his starting point. What is the difference between the angle of inclination on the east side and the angle of inclination on the west side?

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