9-1 Notes. Learning Goal: What are trigonometric ratios and how can we use them to solve for a side? Flashback!

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1 9-1 Notes Learning Goal: What are trigonometric ratios and how can we use them to solve for a side? Example 1) Solve for the missing side in the right triangle shown below. What s your thinking? Flashback! Find ( ) given ( ) The function is acting on the input to give us an output. Input Output In this unit, we will be discussing three new functions that stem from special relationships within right triangles, so we call then trigonometric functions. Can you find them on your calculator? These functions are special because they only act on angle measures and their values are found by using specific ratios. These ratios are formed by using the lengths of the right triangles. ( ) ( ) where x is the measure of an angle ( ) ( ) where x is the measure of an angle ( ) ( ) where x is the measure of an angle These functions will help us solve for missing angles and missing sides of right triangles when Pythagorean Theorem can t help us. Transfer your knowledge If we see cos(60) then we know that the function is acting on to give us some type of output which we call the VALUE of the function.

2 What will be the output/ VALUE of the Trig Function? To find the output/ value of a trig function, we use ratios! But first, let s discuss angle-side relationships. Angle-Side Relationships Opposite Side The side from a given angle Adjacent Side The side a given angle * NOT the Hypotenuse The side the Let's Practice 1) What is the length of the side opposite from angle A? 2) What is the length of the side adjacent to angle B? how the sides are labeled here! Where is side a located? 3) What is the length of the hypotenuse? Sides of triangles are sometimes labeled by the angle them! When calculating the values of the functions, we will always use these special ratios! Trigonometric Ratios Or, SOH CAH TOA

3 Let s practice writing trig ratios! 1) The diagram right, shows right triangle XYZ. Write the ratio that represents... a) the sine of X b) the cosine of X c) the tangent of Y d) the cosine of Y Solving for a Side Using Trigonometric Ratios CHECK YOUR CALCULATOR MODE 2) In right triangle ABC, hypotenuse AB=15 and angle A=35º. Find leg length, BC, to the nearest tenth. Why CAN T we use Pythagorean Theorem here?? Let s try another! 3) In right triangle ABC, leg length BC=20 and angle B = 41. Find hypotenuse length BA to the nearest hundredth.

4 SOHCAHTOA PARTNER PLAYTIME Choose who will be partner peanut butter and who will be partner jelly. Work on your column s questions. Each problem is different, but your multiple choice answers should match. If there is a discrepancy between answers, figure out who is right and help your partner correct their mistake. Work on the last 2 problems together Partner Peanut Butter 1 In below, the measure of,,, and. Which ratio represents the sine of? 1) 3) Partner Jelly The diagram below shows right triangle UPC. Which ratio represents the sine of? 1) 3) 2) 4) 2) 4) 2 Which ratio represents the cosine of angle A in the right triangle below? 1) 3) Which ratio represents diagram of? 1) 3) in the accompanying 2) 4) 2) 4) 3 In the accompanying diagram of right triangle ABC,,,, and. What is? 1) 3) The diagram below shows right triangle ABC. Which ratio represents the tangent of? 1) 3) 2) 4) 2) 4) 4 In triangle MCT, the measure of,,, and Which ratio represents the sine of? 1) 3) In, the measure of,, and. Which ratio represents the tangent of? 1) 3) 2) 4) 2) 4)

5 5 As shown in the diagram below, a building casts a 72 foot shadow on the ground when the angle of elevation of the Sun is 40. Partner Peanut Butter Partner Jelly How tall is the building, to the nearest foot? The accompanying diagram shows a ramp 30 feet long leaning against a wall at a construction site. If the ramp forms an angle of 32 with the ground, how high above the ground, to the nearest tenth, is the top of the ramp? 1) 60 2) 46 3) 86 4) 94 1) 15.9 ft 2) 18.7 ft 3) 25.4 ft 4) 56.6 ft 6 As shown in the diagram below, the angle of elevation from a point on the ground to the top of the tree is 34. A tree casts a 25-foot shadow on a sunny day, as shown in the diagram below. If the point is 20 feet from the base of the tree, what is the height of the tree, to the nearest tenth of a foot? 1) ) ) ) 11.2 If the angle of elevation from the tip of the shadow to the top of the tree is 32, what is the height of the tree to the nearest tenth of a foot? 1) ) ) ) 40.0 Peanut Butter Jelly Time! Try these as a team now 7. A support post leans against a wall, making a 70 angle with the ground. The support post is 20 feet above the ground where it leans along the wall. To the nearest tenth of a foot, how far along the ground is the base of the support from the wall?

6 8. In right triangle ABC,,, and. What is the length of the hypotenuse? 1) 2) 3) 8 4) 9. From a point on level ground 25 ft. from the base of a tower, the angle of elevation to the top of the tower is, as shown in the accompanying diagram. Find the height of the tower to the nearest tenth of a foot. 10. A lighthouse is built on the edge of a cliff near the ocean, as shown in the accompanying diagram. From a boat located 200 feet from the base of the cliff, the angle of elevation to the top of the cliff is 18 and the angle of elevation to the top of the lighthouse is 28. What is the height of the lighthouse, x, to the nearest tenth of a foot?

7 9-1 HW Directions: Complete EACH of the following problems; CHECK YOUR ANSWERS!! 1) Using the diagram below find the sine, cosine and tangent ratios of ABC? Sin: Cos: Tan: 2) A ramp is leaning against a wall. The angle the ramp makes with the ground is 18. The length of the ramp is 12 feet. Find the length from the bottom of the ramp to the wall to the nearest tenth. In the diagram below, a window of a house is 15 feet above the ground. A ladder is placed against the house with its base at an angle of 75 with the ground. Determine and state the length of the ladder to the nearest tenth of a foot.

8 5) The tailgate of a truck is 2 ft above the ground. The incline of a ramp used for loading the truck is, as shown below. Find to the nearest tenth of a foot, the length of the ramp. 6) Sketch right triangle DEF with E the right angle. If cos<dfe =, determine the sin<dfe (Leave any radicals in your final answer, NO DECIMALS). 7) The diagram shows a rectangle ABCD with diagonal BD drawn. What is the perimeter of the rectangle (ABCD) to the nearest tenth?

9 9-2 Notes Today s Learning Goal: How can we use trigonometric ratios to solve for a missing angle of a right triangle? Read the following: In right triangle ABC, leg BC=15 and leg AC=20.Find angle A to the nearest degree. *Turn and Talk: How is this example different than examples we had yesterday? Stretch before we go! Think back to Algebra 1. What is the VALUE of x? What property justifies this action? Making Connections How would you solve to get x alone? Sin(x) = Watch out! Common Mistake: x 1 2

10 Back to that problem from before. using Trig to solve for an angle 1) In right triangle ABC, leg BC=15 and leg AC=20. Find angle A to the nearest degree. Important Notation sin -1 cos -1 tan -1 : We read. We read. We read. Tips for Success 1) Always draw a diagram and carefully label 2) There are two descriptions of angles that can be given in problems: a) Angle of Elevation b) Angle of Depression *Angle of elevation and angle of depression are through 3) Always use the inverse of a function to find an in a right triangle. 4) ALWAYS round your answer at the end. The ONLY thing you should round is your FINAL ANSWER 5) Use calculator ALWAYS for calculators

11 2) One more together! Standing on the deck at the top of a light house, a person spots a ship off in the distance. The lighthouse is m tall and sits on a cliff. The vertical height from the ship to the lighthouse is 2km. The horizontal distance between the light house and the ship is 5600m. Use the sketch below and determine the angle of elevation from the ship to the lighthouse deck to the nearest hundredth. You re Turn! Complete the rest of today s practice! 3) A piece of lumber leans against a wall. The top of this 40 foot piece of lumber touches a point on the wall that is 36 feet above the ground. Find to the nearest degree the measure of the angle that the lumber makes with the wall. 4) Which equation could be used to find the measure of one acute angle in the right triangle shown below? (1) sin A 4 5 (3) cos B 5 4 (2) tan A 5 4 (4) tan B 4 5

12 5) Ron and Leslie are building a ramp for performing skateboard stunts, as shown in the accompanying diagram. The ramp is 7 feet long and 3 feet high. What is the measure of the angle, x, that the ramp makes with the ground, to the nearest tenth of a degree? 6) The center pole of a tent is 8 feet long, and a side of the tent is 12 feet long as shown in the diagram below. If a right angle is formed where the center pole meets the ground, what is the measure of angle A to the nearest degree? (1) 34 (3) 48 (2) 42 (4) 56 Follow Up: Using your answer from above, what is the distance of A to the center pole in the ground in 2 different ways. Round to the nearest whole number.

13 9-2 Homework 1) In the diagram of ABC shown below, BC 10 and AB 16. 2) To the nearest tenth of a degree, what is the measure of the largest acute angle in the triangle? (1) 32.0 (3) 51.3 (2) 38.7 (4) ) A person standing on level ground is 2,000 feet away from the foot of a 420-foot-tall building, as shown in the accompanying diagram. To the nearest degree, what is the value of x? 3) A man who is 5 feet 9 inches tall casts a shadow of 8 feet 6 inches. Assuming that the man is standing perpendicular to the ground, what is the angle of elevation from the end of the shadow to the top of the man s head, to the nearest tenth of a degree? (1) 34.1 (2) 34.5 (3) 42.6 (4) 55.9

14 4) Tyler, who is eye level is 1.5 m above the ground, stands 30m from a tree. The tree is 10.5m tall. What is the angle of elevation to a bird at the top of the tree from Tyler s viewpoint to the nearest tenth of a degree? (Fill in the diagram below to help you!) Follow Up: Without calculating, what is the angle of depression? How do you know? 5) Ana is standing in a building and looking out of a window at a tree. The tree is 20 feet away from Ana. Ana s line of sight to the top of the tree creates a 42 o angle of elevation, and her line of sight to the base of the tree creates a 31 o angle of depression. Hint: Find the two right triangles and trace over them! What is the height of the tree to the nearest foot?

15 Let s make some connections! 9-3 Notes Today s Learning Goal: Making connections with triangle relationships. With your shoulder Buddy! Read through the Regents question below. 1. Identify any concepts, you recognize from this unit or previous units. 2. Jot down any initial thoughts on how to solve the problem! There are multiple ways! The diagram below shows two similar triangles. If, what is the value of to the nearest tenth? Together! What useful information does tell us? What can we conclude about the corresponding sides of the triangles if we are told they are similar? Let s re-draw and solve!

16 Practice Time! With a partner, work through the following questions; make sure you check in with the answer key after you ve completed each page. You will have a check out quiz at the end of the period, so ask questions and pay attention to details! 1. When in doubt draw it out! 2. Why are the parallel segments important? What is the relationship between the triangles? Re-draw! 3.

17 4. a) Which angles must be are congruent? b) Fill in the blanks: 5. Hints! What do we know about the triangles? Remember! The Order of the letters in each triangle name will tell us which angles are congruent!

18 6. In the accompanying diagram, the base of a 15-foot ladder rests on the ground 4 feet from a 6-foot fence. a) If the ladder touches the top of the fence and the side of a building, what angle, to the nearest degree, does the ladder make with the ground? b) Using the angle found in part a, determine how far the top of the ladder reaches up the side of the building, to the nearest foot. 7. As shown in the diagram below, the angle of elevation from a point on the ground to the top of the tree is 34. If the point is 20 feet the nearest tenth of a foot? from the base of the tree, what is the height of the tree, to

19 9-3 HW Complete the following problems show all work for solutions. 1. As shown in the diagram below, a ship is heading directly toward a lighthouse whose beacon is 125 feet above sea level. At the first sighting, point A, the angle of elevation from the ship to the light was 7. A short time later, at point D, the angle of elevation was 16. To the nearest foot, determine and state how far the ship traveled from point A to point D. 2. Bert and Ernie are racing to get to the island from their location on the beach. Since Bert is a faster swimmer, he told Ernie he would start from a little further away. How much further does Bert have to swim than Ernie to the nearest 10 th of a meter?

20 3. In right triangle ABC,,, and. What is the length of the hypotenuse? 1) 2) 3) 8 4) 4. From a point on level ground 25 ft. from the base of a tower, the angle of elevation to the top of the tower is, as shown in the accompanying diagram. Find the height of the tower to the nearest tenth of a foot. 5. A lighthouse is built on the edge of a cliff near the ocean, as shown in the accompanying diagram. From a boat located 200 feet from the base of the cliff, the angle of elevation to the top of the cliff is 18 and the angle of elevation to the top of the lighthouse is 28. What is the height of the lighthouse, x, to the nearest tenth of a foot?

21 9-4 Notes Today s Learning Goal: How can solve for sides or angles in non-right triangles? Warm-Up: In triangle ABC, BA = 72, m<b = 13 o, m<a = 38 o and m<c = Solve for the measure of side BC. a) Can we use SOHCAHTOA to solve for BC? Explain b) What makes this problem different than problems we have seen in this unit? Law of Sines When do we use the law of sines? We use the law of sines when we are given angles and sides, including the piece we are trying to find, in type of triangle

22 What is the law of sines? Law of Sines: ( ) ( ) Things to remember: The ratio is always set up with the side over the sine of the angle Once you find a missing part of a triangle, add it to your diagram! Back to the warm-up! 1. In triangle ABC, BA = 72, m<b = 13 o, and m<a = 38 o. Solve for the measure of side BC to the nearest foot. Steps: 1. Set up the law of sines with your labels 2. Plug in the information you are given 3. Cross multiply 4. Solve for the missing variable 5. Don t forget to add your answer to the diagram after you find it!

23 2. In triangle PQR, PQ = 41, PR = 28 and m<pqr = 39 o. Solve for R to the nearest tenth. What makes this problem different? This means we have to use What is the measure of the largest angle in triangle PQR? What s the longest side?

24 Let s Try one more! 3. Find all of the unknown side lengths to the nearest tenth. Solve for side y: Wait!!! Do we need any other information before we solve for x? Now Solve for x:

25 PRACTICE TIME 4. In the triangle DEF, angle D is 72 o, side EF is 12.1cm, and side ED is 4.5 cm. Find angle F to the nearest hundredth of a degree. 5. John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle of elevation to the top of the tree is 33. This particular tree grows at an angle of 83 with respect to the ground rather than vertically (90 ). How tall is the tree, to the nearest tenth of a foot?

26 6. Solve for each of the missing sides in the triangle shown below to the nearest tenth. 7. Meredith, Ana, and Chrissy are camping in their tents. The distance between Meredith and Ana is 153 feet, the distance between Meredith and Chrissy is 201 feet, and the distance between Ana and Chrissy is 175 feet. The angle from Chrissy to Meredith and Ana is 34. What is the angle from Ana? Round to the nearest tenth.

27 9-4 Homework 1. a. Find side AC, to the nearest whole number. b. Find side CB to the nearest whole number. 2. In triangle ABC, AC = 5, BC = 7, m<cab = 48, as shown in the diagram. Find m<cba giving your answer correct to the nearest degree. C 5 48 A 7 diagram not to scale B

28 3. The following diagram shows a triangle ABC, where BC = 5 cm, <B= 60, <C= 40. Calculate AB to the nearest whole number. A B cm C 4. Cathy wants to determine the height of the flagpole shown in the diagram below. She uses a survey instrument to measure the angle of elevation to the top of the flagpole, and determines it to be She walks 8 meters closer and determines the new measure of the angle of elevation to be At each measurement, the survey instrument is 1.7 meters above the ground. Determine and state, to the nearest tenth of a meter, the height of the flagpole.

29 Break it down. a) Solve for and add into diagram: i) m<acd ii) Solve for m<cad iii) Solve for m<bac b) Use Law of Sines to solve for AD (keep a few decimals) c) Use Sohcahtoa to solve for AB (keep a few decimals) d) What is the height of the flagpole to the nearest tenth?

30 9-5 Notes Learning Goal: How can we use the Law of Cosines to solve for missing sides and angles in non-right triangles? Warm-Up: Two hot air balloons approach a landing field. One is 12 meters from the landing point and the other is 17 meters from the landing point. The angle between the balloons is 70. How far apart are the two balloons? How is this question different from questions we ve worked with yesterday? Can we use The Pythagorean Theorem? Explain. Can we use Law of Sines? If yes, use it. If not, explain what you other information you would need in order to use it. Law of Cosines When do we use the law of Cosines? We use the law of cosines when we are given sides and angle, including the piece we are trying to find. Law of Cosines ( ) Things to remember: The side by itself must be the side from the angle with Cosine. Plug the entire right side of the equation in and find the solution Square root your answer to get the exact value of the side!

31 Now, let s Solve the Warm-up Question! 1. Two hot air balloons approach a landing field. One is 12 meters from the landing point and the other is 17 meters from the landing point. The angle between the balloons is 70. How far apart are the two balloons to the nearest meter? Your calculator should look like: 2. After a hurricane, the small tree in my neighbor s yard was leaning. To keep it from falling down, we nailed a 6-foot strap into the ground 4 feet from the base of the tree. We attached the strap to the tree 3½ feet above the ground. At what angle was the tree leaning away from the ground, to the nearest degree? How is this question and question #1 different? How are they similar?

32 You Try! 3. Consider the triangle to the right. a. Solve for BC to the nearest whole number: b. c. Using Law of Cosines, solve for angle C to the nearest degree: *Check in with key! Then, continue to practice. 4. The diagram shows the plan of a playground with dimensions as shown. Calculate the length BC to the nearest tenth of a meter. C 48 m 117º A 57 m B

33 5. A cross country race is taking place, and Matt is trying to figure out the distance of the course. He receives a map of the course, which is in the shape of a triangle. However, the map he receives was misprinted. He only has 5 minutes until the race starts, and he wants to know how far he will have to run in order to complete the course. The map is shown below with all of the printed information. Hint: Do we need any other information before solving for the missing side? a. Find the missing side of the triangular trail that was misprinted to the nearest meter. b. Calculate the total distance Matt with run during his race. 2. Find the largest angle to the nearest tenth of a degree of a triangle whose sides are 9, 12, 18.

34 9-5 Homework 1. Determine the length of AC to the nearest hundredth of a cm. 2. Given the following measures in DEF, e = 3, f = 7, d = 9, find the measure of the smallest angle to the nearest tenth of a degree. 3. The XYZ has XZ =8 inches, YZ = x, and XY = 6 inches as shown below. The perimeter of XYZ is 18in. Find the measure of angle X to the nearest 100 th of a degree. 4. The lengths of two adjacent sides of a parallelogram are 12cm and 15cm. The measurement of the angle in between these two sides is 156. Using Trig, find the length of the longer diagonal to the nearest centimeter.

35 5. As shown in the diagram below, fire-tracking station A is 100 miles due west of fire-tracking station B. A forest fire is spotted at F, on a bearing 47 northeast of station A and 15 northeast of station B. Determine, to the nearest 10 th of a mile, the distance the fire is from both station A and station B. [N represents due north.] 6. Solve for a, b, and c. Show all of your work for each! Round to the nearest 10 th.

36 9-6 Notes Today s Learning Goal: Which Trigonometric rules can we use now? Warm-Up: Examine the following problems displayed below. Match the problem with the appropriate Trigonometric rule that would be used to solve it! DO NOT SOLVE THEM!! 1. Use the following examples of non-right triangles to fill in the chart below: Example A) Solve for ZY Example B) Solve for AC Number of Sides Involved in Problem Number of Angles Involved in Problem Which Rule you would use: WRITE THE RULE! 2. Use the following examples of right triangles to fill in the chart below: State the rule/theorem you would use to solve for the missing part? Example A) Example B) Example C)

37 Jump into it! Practice time! 1. Using the following diagram: a. To solve for side LM, which law do you need to use here? How do you know? b. Solve for. c. Solve for side length LM in the triangle above. Round your answer to the nearest hundredth. 2. Two airplanes leave an airport, and the angle between their flight paths is 40. An hour later, one plane has traveled 300 miles while the other has traveled 200 miles. How far apart are the planes at this time, to the nearest hundredth of a mile? a. Sketch the diagram! b. Which law will you use? How do you know? c. Solve it!

38 3. In triangle ABC, a = 55, c = 20, and m A = 110. Find the measure of C to the nearest degree 4. Three friends are camping in the woods, Bert, Ernie and Elmo. They each have their own tent and the tents are set up in a triangle. Bert and Ernie are 10m apart, Bert and Elmo are 12 m apart, and Ernie and Elmo are 15 m apart. Find the angle formed at Elmo s tent, to the nearest degree. 5. A canoe race is to be run over a triangular course marked by buoys A, B and C. The distance between A and B is 100 yards, the distance between B and C is 160 yards, and the distance between C and A is 220 yards. Find, to the nearest degree, m ABC. 5.

39 6. The center pole of a tent is 8 feet long, and a side of the tent is 12 feet long as shown in the diagram below. If a right angle is formed where the center pole meets the ground, what is the measure of angle A to the nearest degree? 7. In, the measure of,, and. Which ratio represents the tangent of? 1) 2) 3) 4) 8. In right triangle ABC shown below,,, and. Which equation is not correct? 1) 2) 3) 4)

40 9-7 Notes Lesson 9-7: Re-writing Trigonometric Expressions-Cofunctions Today s Aim: How can we use relationships within right triangles to help us re-write Trigonometric ratios? Le Re-activate our knowledge What do we call these different parts of a trig function? The or of the trig function. The ( function. ) of the trig Often, mathematicians will use Greek letters to identify an angle in a right triangle. Theta Try writing some here: Group Exploration READ and answer the following questions in your groups: 1. What type of triangle is shown here? How do you know? 2. Using your answer from Question # 1 and your knowledge of the interior angles of triangles, what MUST be the sum of Therefore the two non-right angles in a right triangle must always be. 3. Write the indicated trig ratios using the sides of the triangle shown above : = 4. What do you notice about these values? hmm..

41 4b. Will this be true for ALL right triangles? Try another ( Triangle ABC) = Let s summarize as a class: The VALUE of sine of an acute angle is equal to the _ complement. of its Symbols: Example: The VALUE of cosine of an acute angle is equal to the complement. of its Symbols: Example In reverse: If two acute angles are complementary, then the VALUE of the cosine of one of the angles is equal to the VALUE of the sine of the other angle. Symbols: If = 90, then sin(a) = cos(b) We call Sine and Cosine. Let s try! 1. Write each expression as a function of an acute angle. a. sin 80 b. cos 36

42 2. Each equation contains the measures of two acute angles. Find a value of θ for which the statement is true. Remember! In cofunctions, VALUES are equal and ANGLES are complements. 1) sin 10 = cos θ 2) sin θ = cos 2θ 3. In right triangle ABC with the right angle at C, sin A = 2x and cos B = 4x 0.7. Determine and state the value of x. Explain your answer. You re Turn! 4) Write the expression in terms of sine and/or cosine. A) sin 7 B) cos 31 5) Which ratios are equal to ½? Select all that apply: a. sinl b. cosl c. sinj d. cosj 6) Write sin(30 ) in terms of its co-function.

43 7) Math-Hoo uses the equation sin49 = 1 to find BC. His cousin Hal uses the equation cos41 = 1 to find BC. Who is correct? Explain. 8) Fill it in: 7) In, the complement of is. Which statement is always true? 1) 2) 3) 4) 8) In right triangle ABC with the right angle at C, sin A = 2x + 5 and cos B = 4x 15. Solve for x. Explain your solution.

44 9-7 Homework 1) Find: sin J, sin K, cos J, and cos K. Write each answer as a ratio. 2) Write the expression in terms of sine or cosine. a) sin 22 = cos b) cos 56 c) cos 15 3) Find the value of x and y. Round to the nearest tenth.

45 4) The angle of depression is 11 from the bottom of a boat to a deep sea diver at a depth of 120 feet. Find the distance x the diver must swim up to the boat to the nearest foot. 5) In, where is a right angle,. What is? 6) Find the value of R that will make the equation true when. Explain your answer. 7) In right triangle ABC with the right angle at C, sin A = 3x + 5 and cos B = x 40. Solve for x.