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Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry 1 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Trigonometry Exploration 1 (cont’d) Use the the triangles provided for you and the two new ones that you created to fill out the charts below. You will need to use a protractor to measure the angles. How do the side lengths of triangles DEF and GHI compare to those of triangle ABC? How do the angles of triangles DEF and GHI compare to those of triangle ABC? 2 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Chart 2 Calculate the following: Ratio Fraction Decimal Fraction Decimal What do you notice about the results found in these charts? a/c b/c a/b Ratio How would you explain what you have found to a classmate who was absent today? d/f e/f d/e Ratio Fraction Decimal g/i Compare your answers with a classmate. Are there any differences? Why? h/i g/h If I created another triangle and labelled it JKL (see diagram below) and this triangle was a different size but exactly the same shape as ABC, what would the following ratios be? j/l = k/l = j/k = 3 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Trigonometry Exploration 2 Using a ruler and a protractor and the grid on the reverse, create a triangle that has the following characteristics: (You will have to measure, you will not be able to “count blocks”.) Angle A = ___________ Angle B = ___________ Side b = ____________ Fill out the missing information once you have finished carefully drawing the triangle. You will need to carefully measure the angles and sides. Record your answers to the nearest millimetre. Side a = Side b = Side c = Complete the ratios. Write your answer as a fraction, then also as a decimal. a/c = b/c = a/b = Now find another person who had the same triangle. It will have exactly the same shape, but might be a different size. 4 Foundations oundations of Math 10 Chapter 3 – Right Triangle Trigonometry 5 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Now with your triangle partners, compare the ratios you calculated. Come to some agreement for the first 3 decimal values for each ratio. Record your answers below. a/c = b/c = a/b = Together with your partners, answer the following questions: 1. What determines the shape of a triangle? Explain your answer. 2. What have you noticed about the way the sides of identically-shaped triangles compare to oneanother even though they are different sizes? 3. Complete these sentences with the words in the table below. A word may be used more than once or not at all. Triangles that are exactly the same shape (have the _______________) but have different ______________ are called ______________ triangles. If two triangles are __________, then the _____________ of their corresponding sides will be _____________. In trigonometry, we can make use of these relationships to help us find missing information about triangles, provided we have information about a triangle that is _______________ to the one we are looking at. similar ratios same angles side lengths equal 6 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Class Summary As a class, we are going to fill out the table below. This will help us to find information about triangles that are similar to the ones in the table. Angle A a/c b/c a/b 15 o 30 o 45 o 60 o For example: Find the value of x. 18 cm x 30° 7 cm 60° x 7 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry DAY 3 – SINE, COSINE, AND TANGENT 8 Foundations oundations of Math 10 Chapter 3 – Right Triangle Trigonometry We can use what we have found out about triangles to solve many problems. It is an entirely new area of mathematics to you and it has different rules and operations than what you are used to with arithmetic and algebra. In the past few lessons we learned that triangles that are exactly the same shape (same angles) but have different sizes are called similar triangles. triangles. If two triangles are similar then the ratios of their corresponding sides will be equal. In special special cases, when one angle of the triangle is a right angle (90 ) then the triangle is called a right triangle. triangle We can use ratios of similar triangles to find missing lengths or angles, just as we did in the beginning of the activity today. What happens if I wanted to find information about a triangle with an angle of 11 , or 23 , or some other angle? It would be very time consuming to draw and measure each triangle and then calculate the missing side or angle. In the the past there used to be very long trig tables (like we made, but longer) and you could look up the ratios. Let’s take another look…… If we look at the labelled triangle we can see that a is A opposite angle A, and that b is adjacent (next) to angle b A and c is the longest side (across from the right angle) c and it is called the hypotenuse. C B a 9 Foundations oundations of Math 10 Chapter 3 – Right Triangle Trigonometry This means if we look from angle A, the ratios can be expressed as: This is referred to as the sine of angle A, or . This means that if I had a triangle with an angle of 40◦ 40 and I wanted to find out what the ratio of the opposite side to the hypotenuse was (or ) I would look up or calculate This is referred to as the cosine of angle A, or . This is referred to as the tangent of angle A, or . . Let’s take another look at your class trig table. Use your calculator to find each ratio. Does it match your table from the last lesson? Angle A a/c b/c a/b 15 30 45 60 90 10 Foundations oundations of Math 10 Chapter 3 – Right Triangle Trigonometry Now we can use the sine, cosine, and tangent ratios to find missing sides and angles in right triangles. Ex.. Find the measure of x. Look at the angle. What sides do we have? 56 cm ______________ and ______________. Which trigonometric ratio is this? Set up the ratio. x 33° Solve for x.. Ex. Find . A 35 B 16 23 C Ex.. Find the measure of t. U 23◦ t= S s = 15 m T Assignment: With your partner, do the questions starting on page 120 #1,2, 4, 6bd, 8. Assignment: 11 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry DAY 4 – SOLVING RIGHT TRIANGLES (I) To “solve” a triangle means to find all missing sides and all missing angles. We have learned how to find missing sides using the appropriate ratio, but there are some other methods for solving triangles that we can use. Ex. Find the missing side. 16 cm 20 cm Ex. Find the indicated angle. Ө 17 m 11 m If we look from the angle we see we have two sides; the ____________________ and the ______________________. We know that this represents the ________________ ratio. So we can set up the equation: If we had that ratio in our table we could look up the angle (we could use the big table we made – try that first). In the past, we would have had a book that we could look up the ratios in. But today we will use a calculator. We already know the ratio, but we need the angle so we have to use a reverse function, or inverse function. Find the inverse function for the ___________ ratio on your calculator. Enter the inverse function, then the ratio and hit enter. The calculator “looks up” the angle that goes with that ratio and shows the resulting angle on the screen. (Note: On some calculators you will have to type the ratio first, then the inverse function.) Show what you did here: 12 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Ex. Find the indicated angle. 21 in 15 in Ө Assignment: Do the questions starting on page 120 #3, 5, 6ac, 7, 12, 13 , and 16. Activity: Design a poster/study sheet that shows all you know about right triangles so far. 13 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry DAY 5 – SOLVING RIGHT TRIANGLES (II) To “solve” a triangle means to find all missing sides and all missing angles. We have explored a number of methods to solve right triangles. What are these? 1. ____________________________________________ 2. ____________________________________________ 3. ____________________________________________ Using some or all of these methods, solve the triangles below. A Q 17 m 43 cm 11 m C B 25⁰ S R 14 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry ANGLE OF ELEVATION AND DEPRESSION PROBLEMS When looking up, the angle of elevation is the angle your line of sight makes with the horizontal. angle of elevation horizontal angle of depression When looking down, the angle of depression is the angle your line of sight makes with the horizontal. Ex. A guy wire runs from a point on the ground 5 metres from the base of a flag pole to its top. The angle of elevation of the wire is 65°. How long is the wire? Answer to the nearest tenth of a metre. wire length, w flag pole 65° 5m ground 15 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Ex. A man standing on top of a building looks down to see the building’s shadow. The angle of depression is 35⁰. What is the length of the building’s shadow if the height of the building is 40 ft? Answer to the nearest tenth of a foot. Ex. A farmer wants to determine the height of a tree he needs to chop down. From a distance 10m from the tree he measures the angle of elevation to be 65⁰. What is the height of the tree? 16 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Name: _________________________ Group Members: _______________________ Height of a Flagpole – Trigonometry Challenge Part A: Your challenge today is to use trigonometry to estimate the height of the flagpole at the front of the school. You may use only the tools below (and a calculator). (You may not climb the flagpole, nor any other part of the school). Show all your methods and calculations below. 17 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Part B: Imagine someone has taken away your protractor. Can you think of a way to find the height of the flagpole using only the metre stick, string and a tape measure? Try it – explain your method and show all work below. 18 Foundations of Math 10 Chapter 3 – Right Triangle Trigonometry Answer the following questions in full sentences. What strategy did your group employ to find the height? Why did you decide to use this strategy? Was the strategy effective? Is there anything that would have improved it? Describe in your own words what you learned about trigonometry during this activity. 19