Foundations of Math II Unit 4: Trigonometry Academics High School Mathematics
4.1 Warm Up 1) a) Accurately draw a ramp which forms a 14 angle with the ground, using the grid below. b) Find the height of a support board which could be used to make your ramp, and the distance of the support board from the beginning of your ramp. c) Draw another ramp which forms a 14 angle with the ground, and give its measurements as in part (b). d) What do you notice? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 2
4.1 Lesson Handout 1. Jill is building a ramp. She knows that she must place the support board 142 in. from the base of the ramp. The ramp must make a 22 angle with the ground. She needs to figure out how high to make the support board. Jill draws a picture of the situation and asks Bill to help her solve for q in the triangle shown. Bill says, If you tell me the slope of a 22 line, a line which forms a 22 angle with the x-axis, I can tell you what q is. B q A 22 142 C a) Help Jill accurately find the slope of a 22 line. b) How will Bill find q? c) Repeat parts (a) and (b) if A = 44. Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 3
4.1 Practice Now Jill wants to build another ramp. This time the ramp must make a 40 angle with the ground and she has a support board that is 10 in. high. She needs to know how far from the base of the ramp to place the support board. Jill draws a picture of the situation and asks Bill to help her solve for x in the triangle shown below, which is not drawn to scale. Bill says, If you tell me the slope of a 40 line, I can tell you what x is. B 10 in. 40 A x C a) Help Jill accurately find the slope of a 40 line. b) How will Bill now find x? Find x. c) Repeat parts (a) and (b) if A = 62. d) Repeat parts (a) and (b) if A = 9. e) Repeat parts (a) and (b) if A = 22 f) Bill says, This is fun! Let s do a few more! Jill says, I am getting tired of all this work, Bill. There must be an easier way to find slopes. What do you think? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 4
4.2 Practice Tangent Ratio 1) Find tan A 5 C B 12 13 A a) Mark A. b) What is the length of the support board (opposite side)? c) What is the length of the ramp (hypotenuse)? d) What is the distance from the support board to the base of the ramp (adjacent side)? e) Write the tangent ratio for A. f) Change the ratio to decimal form. Use the table to find the value of A. 2) Find tan B. a) Redraw ABC so that BC is the base of the ramp and AC is the support board. Mark B. b) What is the length of the support board (opposite side)? c) What is the length of the ramp (hypotenuse)? d) What is the distance from the support board to the base of the ramp (adjacent side)? e) Write the tangent ratio for B. f) Change the ratio to decimal form. Use the table to find the value of B. How else can you find the measure of B? 3) Find the missing measurements in each triangle below. All measurements are given in centimeters. 4) Jill says to Bill, I know the answer to this one without having to write anything down. What do you think Jill means? 5
4.2 Homework Tangent Ratio 1) Find tan N a) Mark N. b) What is the length of the support board (opposite side)? c) What is the length of the ramp (hypotenuse)? (Hint: Use the Pythagorean Theorem.) d) What is the distance from the support board to the base of the ramp (adjacent side)? e) Write the tangent ratio for N. f) Change the ratio to decimal form. Use the table to find the value of N. 2) Find tan M. a) Redraw MNP so that MP is the base of the ramp and NP is the support board. Mark M. b) What is the length of the support board (opposite side)? c) What is the length of the ramp (hypotenuse)? d) What is the distance from the support board to the base of the ramp (adjacent side)? e) Write the tangent ratio for M. f) Change the ratio to decimal form. Use the table to find the measure of M. How else can you find the value of M? 3) DM = 5) x y 7 40 y x 4) TR = AT = m T= 6) q = y = q y 20 28 6
4.3 Applications of the Tangent Ratio Lesson Handout Example 1 Jenna goes on an exciting airplane ride. She takes off at a 25 angle and continues flying in a perfectly straight path until she is directly over her house, as shown. She notices that her altitude when directly over her house is 3,200 feet. What distance has she flown? Example 2 Carl decides to use what he has learned about trigonometry to help him find the height of his favorite tree. At a certain time of day, he measures the tree s shadow with a tape measure and finds that it is 31 feet long. Then he measures the angle of elevation to the sun using a clinometer and finds that the sun s rays are striking at a 62 angle with the ground. (The angle that the sun s rays make as they strike an object determines the length of the object s shadow.) Use the information and what you know about trigonometry to calculate the height of the tree. 7
Example 3 Erica is standing on one side of a canyon, and her friend Sasha is standing directly across the canyon from her on the other side. They want to know how wide the canyon is. Erica marks her spot and then walks 10 yards along the canyon edge and looks back at Sasha. The angle of her line of sight to Sasha and the path she just walked is 72. Draw a sketch that illustrates this situation. What is the approximate width of the canyon? Example 4 8
4.3 Applications of the Tangent Ratio Practice 1) 2) To see the top of a building 1000 feet away, you look up 28 from the horizontal. What is the height of the building? 3) A guy wire is anchored 12 feet from the base of a pole. The wire makes a 62 angle with the ground. How long is the wire? 4) An evergreen tree is supported by a wire extending from 1.5 feet below the top of the tree to a stake in the ground that is 15 feet from the base of the tree. The wire forms a 58 angle with the ground. How tall is the tree? 9
4.3 Applications of the Tangent Ratio Homework 1) To the nearest tenth of a foot, how tall is a building 100 feet away (d = 100) if the top of the building is sighted at a 20 angle (n = 20 )? 2) If an object is dropped from the top of the leaning tower of Pisa, it will land about 13 feet from the base of the tower. The tower leans at an angle of approximately 86. How far did the object drop? d 86 13 ft. 3) A ramp was built by the loading dock of a building. The height of the loading dock platform is 7 feet. Determine the length of the ramp if it makes a 38 angle with the ground. (Draw a picture!) 4) A jet airplane begins a steady climb of 15 and flies for two ground miles. What was its change in altitude? 10
4.4 Warm Up 1) Jenna goes on another exciting airplane ride. She takes off at a 35 angle and continues flying in a perfectly straight path for five miles. She discovers that she is directly over her house. 5 mi 35 a) How far is her house from the airport? b) What is her altitude? Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 11
4.4 Trigonometric Ratios Lesson Handout The word trigonometry comes from 2 Greek words, trigon, meaning triangle, and metron, meaning measure. The study of trigonometry involves triangle measurement. We will be studying basic Identifying Sides A Hypotenuse side b c Opposite of angle A Opposite of angle B Adjacent to Angle A Adjacent to Angle B C a B The trig ratios we will be studying are,, and The ratios for each function: Sin = Cos = Tan = Setting up Ratios A 3 5 Sin A = Cos A = Sin B = Cos B = C 4 B Tan A = Tan B = X 12 13 Sin X = Cos X = Sin Y = Cos Y = Tan X = Tan Y = Z 5 Y 12
Using a Calculator Sin 39 = Cos 58 = Tan 85 = Sin 30 = Solving for a Side 1. 2. Y X 10 38 12 85 3. 4. z 70 45 A 34 50 13
4.4 Trigonometric Ratios Practice 1. Find sin A. 5 C B 12 13 A 1. Mark A. 2. Label the sides in relation to A (opp, adj, hyp) 3. Circle the sides that are needed to find sin (opp, hyp) 4. Write the sin ratio for A 5. Change the ratio to decimal form. Sin A = 2. Find BC B 50 x C 17 A 1. Mark angle with the given value. 2. Label the sides in relation to the given angle (opp, adj, hyp) 3. Circle the known relationships. (adj, hyp) 4. Decide which trig function uses these two relationships. 5. Write the trig equation used to solve this problem. 6. Solve the equation. BC = 3. Find the measure of each side indicated. Round your answer to the nearest tenth. a. b. 4. Suppose you re flying a kite, and it gets caught at the top of the tree. You ve let out all 100 feet of string for the kite, and the angle that the string makes with the ground is 75 degrees. Instead of worrying about how to get your kite back, you wonder. How tall is that tree? h 100 ft 75 14
4.4 Trigonometric Ratios Homework 1. Find each of the trig ratios for the triangle at the right: sin A = sin B = cos A = cos B = tan A = tan B = 2. Find the measure of each side indicated. Round to the nearest tenth. a. b. c. d. 3. Solve the following triangles. Round your answer to the nearest tenth. a. b. x z 8 z x 5 50 y 36 y 15
4. You are in charge of ordering a new rope for the flagpole. The rope needs to be twice the height of the flagpole. To find out what length of rope is needed, you observe that the pole casts a shadow 12 meters long on the ground. The angle between the sun s rays and the ground is 37. How tall is the pole? How much rope do you need? h 37 12 m 5. A damsel is in distress and is being held captive in a tower. Her knight in shining armor is on the ground below with a ladder. The knight leans the ladder against the tower. When the knight stands 15 feet from the base of the tower and looks up at his precious damsel, the angle between the ladder and the ground is 60. How long does the ladder have to be in order to reach the window? 6. The tailgate of a moving van is 3.5 feet above the ground. A loading ramp is attached to the rear of the van. The angle that the ramp makes with the ground is 10. Find the length of the loading ramp to the nearest tenth of a foot. 16
4.5 Warm Up 1) Find the measure of T and G T 4 3 G R Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 17
4.5 Inverse Trigonometric Ratios Practice 1. Find each angle measure to the nearest degree a. tan A = 2.0503 b. cos Z = 0.1219 c. sin U = 0.8746 2. Find the measure of the indicated angle to the nearest degree. a. b. c. d. 18
4.5 Inverse Trigonometric Ratios Homework 1. Find each angle measure to the nearest degree a. tan Y = 0.6494 b. cos V = 0.6820 c. sin C = 0.2756 2. Find the measure of the indicated angle to the nearest degree. a. b. c. d. 3. Solve each triangle. Round answers to the nearest tenth. a. b. 13 10 26 7 19
4.6 Warm Up 1) Jill attaches a rope to the Wilderness Survival Training Program tower, 60 feet above the ground. Her rope is 90 feet from the base of the tower and forms a 34 angle with the ground, as shown below. Bill wants to attach a rope to the tower so that the angle it forms with the ground is twice as large as that of Jill s rope. If Bill s rope is also 90 feet from the base of the tower, how far above the ground should Bill attach his rope? Explain your answer. 60 feet 34 Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii 20
4.6 Applications Find the Missing Angle Practice 1. Two legs of a right triangle are 16 and 48. Find the measure hypotenuse and all the angles. 2. One leg of a right triangle is 14 while the hypotenuse is 38. Find the measure of the other leg and all the angles. 3. The bottom of 24-foot ladder is 6 feet from the building that the ladder is leaning against. In order for the ladder to be set-up safely the angle the ladder makes with the ground cannot exceed 75. Is the ladder set up safely? How do you know? 4. A jet airplane out of Denver, Colorado needs to clear a 1,500 ft mountain 1 mile (5,280 feet) after it takes off. If the plane makes a steady climb after takeoff, what angle does the plane need to take to clear the mountain? 5. The Washington Monument is 555 feet tall. An observer is 300 feet from the base of the monument. If the observer is lying on the ground looking to the top of the monument, find the angle made between the observer s line of sight and the ground. 21
4.6 Applications Find the Missing Angle Homework 1. One leg of a right triangle is 10 while the hypotenuse is 27. Find the measure of the other leg and all the angles. 2. A road rises 10 feet for every 400 feet along the pavement (not the horizontal). What is the measurement of the angle the road forms with the horizontal? 3. A 32-foot ladder leaning against a building touches the side of the building 26 feet above the ground. What is the measurement of the angle formed by the ladder and the ground? 4. A wire anchored to the ground braces a 17-foot pole. The wire is 20 feet long and is attached to the pole 2 feet from the top of the pole. What angle does the wire make with the ground? 5. Margo is flying a kite at the park and realizes that all 500 feet of string are out. She has staked the kite in the ground. If she knows that the kite is 338 feet high, what is the angle that the kite makes with the ground? 22
4.7 Angles of Elevation and Depression Practice 1 At a certain time of day the angle of elevation of the sun is 44. Find the length of the shadow cast by a building 30 meters high. 2 The top of a lighthouse is 120 meters above sea level. The angle of depression from the top of the lighthouse to the ship is 23. How far is the ship from the foot of the lighthouse? 3 A lighthouse is 100 feet tall. The angle of depression from the top of the lighthouse to one boat is 24. The angle of depression to another boat is 31. How far apart are the boats? 4. At a point on the ground 100 ft. from the foot of a flagpole, the angle of elevation of the top of the pole contains a 31 degree angle. Find the height of the flagpole to the nearest foot. 5. From the top of a lighthouse 190 ft. high, the angle of depression of a boat out at sea is 34 degrees. Find to the nearest foot, the distance from the boat to the foot of the lighthouse. 6. Find to the nearest degree the measure of the angle of elevation of the sun if a post 5 ft. high casts a shadow 10 ft. long. 23
4.7 Angles of Elevation and Depression Homework Draw a picture, write a trig ratio equation, rewrite the equation so that it is calculator ready and then solve each problem. Round measures of segments to the nearest tenth and measures of angles to the nearest degree. 1. A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. What is the ladder s angle of elevation? 2. A 50-meter vertical tower is braced with a cable secured at the top of the tower and tied 30 meters from the base. What is the angle of depression from the top of the tower to the point on the ground where the cable is tied? 3. At a point on the ground 50 feet from the foot of a tree, the angle of elevation to the top of the tree is 53. Find the height of the tree. 4. From the top of a lighthouse 210 feet high, the angle of depression of a boat is 27. Find the distance from the boat to the foot of the lighthouse. The lighthouse was built at sea level. 5. Richard is flying a kite. The kite string has an angle of elevation of 57. If Richard is standing 100 feet from the point on the ground directly below the kite, find the length of the kite string. 6. An airplane rises vertically 1000 feet over a horizontal distance of 5280 feet. What is the angle of elevation of the airplane s path? 24
Making a Clinometer Equipment You will need: A clinometer template cut out of card stock. Some sticky tape. A straw. This needs to be straight enough that you can see all the way through. You may need to snip off any bendy bits. Some thread. A washer. Instructions 1. Cut out the card along the dashed lines 2. Cut a length of thread (about 15cm) 3. Tape the thread so that it hangs along the zero line. Make sure that it pivots at the crosshairs. 4. Tie a washer on the end of the thread to make a plumb line. 5. Tape a drinking straw parallel to the 90 line. It should be as close as possible and must not interfere with the plumb line. 6. Your clinometer is ready to use. 25
Name: Period: Date: Clinometer Lab In this lab, you will create and use a clinometer. A clinometer is an instrument that measures the angle between the ground or the observer and a tall object, such as a tree or a building. To be Completed Inside: 1. Decide the roles: PARTNER A : the Pacer: PARTNER B: the Clinometer Person: 2. Get 2 lengths using a ruler, meter stick, or tape measure: Partner A s (The pacer s) foot length: meters Partner B s (The Clinometer person s) height from floor to eyes: meters 3. Make a clinometer 4. Practice using the Clinometer. a. Look straight ahead at an object in the room (keep the Clinometer horizontal). i.what angle of elevation should this be? ii.what angle does the Clinometer give you? 26
b. Look straight up at the ceiling (make the Clinometer perfectly vertical). i.what angle of elevation should this be? ii. What angle does the Clinometer give you? 5. Use the Clinometer to find the height of the wall in our classroom. STEPS: 1. Partner A, use your foot length to pace out and measure the distance from Partner B to the wall. 2. Partner B, use the Clinometer to get the angle. 3. Write in numbers for the 3? marks. 4. Use trig to find the height of the wall. ** (don t forget to add Partner B s height) ** To be completed outside: Name of Object Measured Clinometer Angle Number of Foot lengths to base of building Calculated distance from base of building (meters) 27
Your mission is to use your data and a little trigonometry to find the height of each of the five objects you chose. Draw AND label a picture. Show all of your work/calculations. 1. Calculated Height 2. Calculated Height 3. Calculated Height 28
4. Calculated Height 5. Calculated Height Summarize Write out the step-by-step process you used to calculate the height of your objects. (Just look at you picture and work and state what you did first, then what you did next, etc). Continue your summary on the back of this paper if you need more room. 29
4.8 Homework 7. A person at one end of a 230-foot bridge spots the river s edge directly below the opposite end of the bridge and finds the angle of depression to be 57. How far below the bridge is the river? 8. The angle of elevation from a car to a tower is 32. The tower is 150 ft. tall. How far is the car from the tower? 230 9. A radio tower 200 ft. high casts a shadow 75 ft. long. What is the angle of elevation of the sun? 10. An escalator from the ground floor to the second floor of a department store is 110 ft long and rises 32 ft. vertically. What is the escalator s angle of elevation? 110 32 11. A rescue team 1000 ft. away from the base of a vertical cliff measures the angle of elevation to the top of the cliff to be 70. A climber is stranded on a ledge. The angle of elevation from the rescue team to the ledge is 55. How far is the stranded climber from the top of the cliff? (Hint: Find y and w using trig ratios. Then subtract w from y to find x) 12. A ladder on a fire truck has its base 8 ft. above the ground. The maximum length of the ladder is 100 ft. If the ladder s greatest angle of elevation possible is 70, what is the highest above the ground that it can reach? y x w 100 100 8 30
4.9 Warm Up Archeologists have recently started uncovering remains of Jamestown Fort in Virginia. The fort was in the shape of an isosceles triangle. Unfortunately, one corner has disappeared into the James River. If the remaining complete wall measures 300 feet and the remaining corners measure 46.5 and 87, what was the approximate area of the original fort? How long were the two incomplete walls? Adapted from Discovering Geometry: An Investigative Approach, Key Curriculum Press 2008 31
4.9 Area of a Triangle Practice Find the area of each triangle below. 1. 2. 3. 4. 5. A new homeowner has a triangular-shaped back yard. Two of the three sides measure 53 ft and 42 ft and form an included angle of 135. To determine the amount of fertilizer and grass seed to be purchased, the owner has to know, or at least approximate, the area of the yard. Find the area of the yard to the nearest square foot. 32
4.9 Area of a Triangle Homework Using your knowledge of area of a triangle, right triangle trigonometry, and the Pythagorean Theorem, find the area of each triangle below. Round your final answer to one decimal place. 1. 2. 3. 4. 5. The intersection sof three roads lea\ves a traingular piece of ground in the middle. What is the area of the grassy section? 6. The area of a triangle is 38 square centimeters. AB is 9 centimeters and BC is 14 centimeters. Calculate the size of the acute angle ABC. 33