MBF 3C Foundations for College Mathematics Grade 11 College Mitchell District High School Unit 1 Trigonometry 9 Video Lessons Allow no more than 15 class days for this unit This includes time for review and to write the test. This does NOT include time for days absent, including snow or fog days. You must make sure you catch up on class days missed. Lesson # Lesson Title Practice Questions Date Completed 1 Review of Ratios Work Sheet U1L1 2 Review of Pythagorean Theorem Work Sheet U1L2 3 4 Finding Side Lengths With the Primary Trig Ratios Finding Angles With the Primary Trig Ratios Consolidation Day Practice Solving for sides and Angles in Right Triangles Page 9 #1-3, 5 Page 15 #1-3 Work Sheet Al Gebra Takes over Trigonometry 5 Applying the Primary Trig Ratios Page 9 #4, 6-9 Page 14 #4-11 6 Introduction to Sine Law Page 25 #1, 3, 6, 7 7 Introduction to Cosine Law Page 35#1-3, 5 8 9 Applying Sine Law and Cosine Law Trigonometry Selecting a Strategy Page 42 #3-9 Page 25 #5, 9, 10 Page 36 #4, 8-10 Work sheet U1L9 Test Written on :
MBF3C U1L1 - Review of Solving Ratios Topic : Goal : Review of Solving Ratios To review skills necessary to solve problems with Trigonometry Solving Ratios There are three ways that you can use to successfully solve ratios. They will work in all situations, but some may be easier at times. If you are more comfortable with one way, there is no reason why you can't use that all the time. Comparison Idea - compare two given numbers in a ratio and decide what you need to multiply or divide by to go from one to the other. Use the same number to multiply or divide to get the unknown. Isolation
MBF3C U1L1 - Review of Solving Ratios Cross Multiplication Practice Questions - Handout Page
MBF3C U1L1ws Solving Ratios Solving Proportional Relationships You learned three methods of solving proportional ratios (Comparison, Isolation and Cross Multiplication). Solve each of the following by the method of your choice. Show all your work. 1. 4 9 = 10 2. 5 x 2 = 6 3. 5 x 2 = 2 x 4. 21 27 = x 18 5. 15 21 = 20 y 6. 26 b = 39 9 7. h 108 = 7 18 8. 45 792 = 70 w 9. 16 120 = j 15 10. 350 p = 1050 60 11. q 1134 = 27 729 12. 40 65 = z 104 13. 15 y = 40 12 14. y 32 = 16 10 15. 32.5 25 = 97.5 q 16. y 42.3 = 144 56.4 17. 126 k = 14 3 18. 30 x = 16 40
MBF3C U1L1ws Solving Ratios 19-22. For each word problem, write and then solve the proportion to find the answer. Be sure to set it up the correct way and show all work. 19. A 380-cubic-centimeter sample of titanium has a mass of 1710 grams. Find the weight of a titanium sample that has a volume of 532 cubic centimeters. Write and then solve a proportion to find the answer. 20. The Bigtown football team outscored its opponents 5:2 last season. If their opponents scored 38 points, how many points did Bigtown score? 21. In the local coed softball league, the male to female ratio is 6:10. If there are 160 players in the league, how many are male? 22. In a certain desert environment there are a lot of small rodents. There also happen to be a lot of snakes that feed on the rodents. The ratio of rodents to rodent eating snakes is 13 to 3. If there are 4,000 snakes in the area, about how many rodents are there? 1. 22.5 2. 2.4 3. 0.8 4. 14 5. 28 6. 6 7. 42 8. 1232 9. 2 10. 20 11. 42 12. 64 13. 4.5 14. 51.2 15. 75 16. 108 17. 27 18. 75 19. 2394g 20. 95 p 21. 60 men 22. about 17333 rodents
MBF3C U1L2 - Review of Pythagorean Theorem Topic : Review of Pythagorean Theorem Goal : To review the skills necessary for studying trigonometry. The Pythagorean Theorem IMPORTANT - YOU MUST HAVE A RIGHT ANGLE IN YOUR TRIANGLE When you square the legs and add them together, you get the same number as when you square the hypotenuse. Example 1. Finding the Hypotenuse when you know the legs. 12 m 8 m
MBF3C U1L2 - Review of Pythagorean Theorem Example 2. Finding a leg when you already know one leg and the Hypotenuse. 15 ft 8 ft Practice Questions - Handout Page
Pythagoras Worksheet Find the lengths of the unknown sides in the following right-angled triangles 1. 2. 3. 4. 5. 6. Simon Jones, www.satuition.com 1
7. Find the length of the diagonal of the rectangle 8. Find the unknown sides 19cm 9. A 5 metre ladder is positioned 3.2 metres from a building, as shown in the diagram. Will the ladder reach a window which is 4 metres from the ground? 10. A cone is shown. Find the diameter of the cone. Hint: the diameter is twice the length of the radius Simon Jones, www.satuition.com 2
11. Find the perpendicular height of the equilateral triangle shown 12. AB is the diameter of the circle shown. Find the radius of the circle 13. HARDER PROBLEM The diagram shows a cube of side 5cm. Find the length of AB. Hint:- All sides of the cube are 5cm. You will need to use Pythagoras twice: find one side and then use it to find AB Simon Jones, www.satuition.com 3
MBF3C U1L3 - Finding Side Lengths with the Primary Trig Ratios Topic : Goal : Finding Side Lengths with Trigonometry I can use the three Primary Trig Ratios to solve for side lengths in right triangles. Finding Sides with the Primary Trig Ratios Sine = Cosine = Tangent = Example 1. Find the missing side length. 1
MBF3C U1L3 - Finding Side Lengths with the Primary Trig Ratios Example 2. Find the missing side length. Practice Questions - Page 9 #1-3, 5 2
MBF3C U1L4 - Finding Angles with the Primary Trig Ratios Topic : Goal : Using Primary Trig Ratios to Find Angles I can find any angle in a right triangle using the primary trig ratios, as long as I have at least two sides Finding Angles with the Primary Trig Ratios Remember : SOH CAH TOA Example 1. Use your calculator to solve for the angle... a) sin A=0.5673 How can I do that? There's no sina button No but there is an inverse sine button. To solve for an angle you need to say to the calculator... Hey, man I know when I divide the opposite by the adjacent side, I get 0.5673 - can you tell me what the angle must have been? b) tan B=0.5673 1
MBF3C U1L4 - Finding Angles with the Primary Trig Ratios Example 2. Solve for the indicated angle in the given diagrams. Step 1 Step 2 Step 3 Step 4 Circle the angle you are finding. Label each side with an O, A or H depending on its relationship to the angle. Pick two sides with numbers given for the lengths and decide which trig ratio uses those sides with SOH CAH TOA. Set up the appropriate ratio and solve accordingly. Remember when finding an angle you will have to press SHIFT before the ratio button. a) b) 14 in 6 in. 22.3 cm 15 cm c) 25 m 20 m 15 m Practice Questions - Page 15 #1-3 2
Al Jebra Takes Over Trigonometry The Evil Dr. Al Jebra has been hard at work trying to take all the enjoyment out of the world of mathematics. He has stolen some of the world s best math jokes and encoded the punch lines using trigonometry. He thinks no one will care enough to wade through all those trig questions to get them back, but your mission is to prove otherwise. Each of the following questions uses sine, cosine or tangent to solve for missing measures of a right triangle. When you have solved for the variable, write its value (rounded to the nearest whole number) in the appropriate box below. Once completed you will have the key to unlocking the math jokes. Give your key to your teacher to check before deciphering the jokes. Good Luck. The world is counting on you. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Al#Jebra#Takes#Over#Trigonometry#5#Codes# WhywasMr.AgardetainedattheairportonhiswaytoIraq? 18,3443,22,55,47,5,20,34,41,32,34,1529,6818,22,30,58,8,63 43,34,22,20,29,8,529,6854,22,32,18 58,8,5,32,4,47,41,32,58,29,8 Whatdidonemathbooksaytoanothermathbook? 15,29,8 3223,29,32,18,34,454,345818,22,30,3454,7 29,43,820,4,29,23,73,34,54,5. Howis2+2=5likeanobtusetriangle? 32,18,34,722,4,348,29,324,58,63,18,32. Whydidtheboyeathismathhomework? 32,18,3432,34,22,41,18,34,432,29,73,1518,58,5458,32 43,22,52220,58,34,41,3429,6841,22,12,34. Dad,canyouhelpmefindthelowestcommondenominatorinthisproblem please? 15,29,8 3232,34,73,7354,3432,18,34,718,22,30,34,8 32 68,29,47,8,1558,327,34,32584,34,54,34,54,23,34,473,29, 29,12,58,8,6368,29,458,3243,18,34,85843,22,522 23,29,7. Whatdidzerosayto8? 8,58,41,3423,34,73,32 WhathappenedwhenMr.AgarpickedOppositeBeachoverAdjacentSki SlopesforhisMarchBreakvacation? 18,3443,22,52232,22,863,34,8,32
MBF3C U1L5 - Applications of the Primary Trig Ratios Topic : Goal : Applying the Primary Trig Ratios I can use SOH CAH TOA to find missing measurements in word problems involving RIGHT triangles. Applying the Primary Trig Ratios Definitions - Angle of Elevation and Depression angle of depression angle of elevation Angles of Depression always look DOWN on something, where angles of elevation look up. An angle of elevation or depression is always made of up of a diagonal arm and horizontal arm NEVER VERTICAL 1
MBF3C U1L5 - Applications of the Primary Trig Ratios The Five Steps of Solving Trig Word Problems Step 1 Step 2 Step 3 Step 4 Step 5 Recreate the situation with objects if necessary. Draw a diagram if one is not given. Label the triangle with sides O, A and H Use SOH CAH TOA to determine which ratio is needed. Solve for the unknown. Be sure to write a concluding statement. Example 1. Sally stood 75m from the base of the CN Tower and looked up to the tip top. She measured the angle of elevation to the top to be 82 o. If Sally's eyes (from where she took the measurement) are about 1.5m above ground, how tall is the CN Tower? 2
MBF3C U1L5 - Applications of the Primary Trig Ratios Example 2. An airplane is flying at an altitude of 1200ft. The pilot spots an airport at an angle of depression of 15 o. What is the distance he must travel in order to be directly above this airport? (Note he is not landing at this airport) Example 3. Mrs. Brodhagen is fixing the sofit on her house. She has a 20 foot ladder that she places 4 feet from the foundation of the house. What is the angle of elevation between the ground and the ladder? Practice Questions - Page 9 #4, 6-9 Page 14 #4-11 3
MBF3C U1L6 - Introductions to Sine Law Topic : Goal : Using Sine Law for Non-Right Triangles I know how to solve for sides and angles in Non-Right Triangles, using the Law of Sines. Introduction to Sine Law For any ABC, where a, b and c are the sides opposite the angles A, B and C respectively, the sine law states... or What you need to use Sine Law... * One angle side pair with numeric values * One other side (if you are finding an angle) or one other angle if you are finding a side. Example 1. Using Sine Law to find an angle... E 25 m 21 m F 52 o G 1
MBF3C U1L6 - Introductions to Sine Law Example 2. Using Sine Law to find a side... F 72 o 18 cm E 67 o G Example 3. Using Sine Law to solve a Non-Right Triangle... 22 ft Q R 26 o 56 o S Practice Questions - Page 23 #1, 3, 6, 7 2
MBF3C U1L7 - Introduction to Cosine Law Topic : Goal : Using Cosine Law to Solve in Non-Right Triangles I can use Cosine Law to solve for angles and side lengths in right triangles where I am unable to use Sine Law. Introduction to Cosine Law When using Sine Law, we had to have (or be able to quickly find) a side-angle pair with numeric values on both. There are two cases where this is not possible. Case 1 - the Pac Man You have values for two sides and the angle between them, and you need to find the last side in the triangle. Q R S Case 2 - You're Surrounded You have values for all three sides of the triangle and you have to find an angle. S Q R In this case we need to introduce the Law of Cosines... For any ABC, where a, b and c are the sides opposite the angles A, B and C respectively, the Cosine Law states... or 1
MBF3C U1L7 - Introduction to Cosine Law Example 1. Solving for a side length using Cosine Law (the Pac Man). Q R S Example 2. Solving for an angle using Cosine Law (You're Surrounded). Q S R Practice Questions - Page 35 #1-3, 5 2
MBF3C U1L8 - Applying the Sine Law & Cosine Law Topic : Goal : Applying Sine Law and Cosine Law I can set up word problems involving non-right triangles and solve using Sine Law or Cosine Law. Applying Sine Law and Cosine Law How can we tell when to use Sine Law and when to use Cosine Law? Think about these cases... I have an angle and the side across from it with known values - plus one other piece of information. Case 1 Case 2 I have a triangle with the side lengths of all three sides known. I have a pacman situation (two sides with values, plus the angle between them) Case 3 Case 4 I have two angles and the side between them with known values. 1
MBF3C U1L8 - Applying the Sine Law & Cosine Law Example 1. A cottage under construction is to be 12.6 m wide. The two sides of the roof are to be supported by rafters the same length that meet at a 50 0 angle. How long should each rafter be? Example 2. Ships A and B at sea are 15.6 km apart. A port (maked by C) can be seen from the deck of each ship. The angles between the line joining the ships and the line of site to the port are 58 0 and 72 0 respectively. How far is each ship from the port? 2
MBF3C U1L8 - Applying the Sine Law & Cosine Law Example 3. A hockey net is 2m wide. A player shoots from a point where the puck is 3.2 m from one goal post and 4.4 m from the other. Within what angle must he make his shot to hit the net? Practice Questions - Page 42 #3-9 (notice that 3 & 4 don't actually ask you to solve the problem) Extra Application Problems Page 25 #5, 9, 10 Page 36 #4, 8-10 3
MBF3C U1L9 - Trigonometry - Selecting a Strategy Topic : Goal : Selecting Strategies I know that there are various ways to solve for sides and angles in triangles and I can select the best strategy for the situation at hand. Selecting a Strategy - Trigonometry Tool Kit Pythagorean Theorem If you know two sides of a RIGHT triangle you can always find the third with Pythagorean Theorem. Sum of Angles Theorem If you know two angles of ANY triangle you can always find the third by subtracting the other two from 180 0 The Primary Trig Ratios (aka SOH CAH TOA) If you have a RIGHT triangle and... - two sides, you can find an angle - a side and an angle, you can find another side. Sine Law If you have a non-right triangle with an angle-side pair having known values you can use Sine Law to... Cosine Law If you have a non-right triangle with all three sides you can use Cosine Law to find an angle...find a side (if you know a second angle) --- find an angle (if you know a second side) If you have a non-right triangle with two sides and the angle between them, you can use Cosine Law to find the third side.
MBF3C U1L9 - Trigonometry - Selecting a Strategy What do we use? Examine each situation and decide which tool from your tool kit will best suit the situation.
MBF3C U1L9 - Trigonometry - Selecting a Strategy Practice Questions - Work Sheet