PreCalculus 4/10/13 Obj: Midterm Review

Transcription

1 PreCalculus 4/10/13 Obj: Midterm Review Agenda 1. Bell Ringer: None 2. #35, 72 Parking lot 37, 39, Homework Requests: Few minutes on Worksheet 4. Exit Ticket: In Class Exam Review Homework: Study for Midterm Exam Announcements: 30 th Week Exam 4/11

2 Exit Ticket: Pg 368 #6, 10 also find the measure of the angle. Pg 368 #31, 33, 35,39, 49, 51 Pg 358 #4, 6, 14, 22, 26, 30, Pg 346 #27, 32,33, 63, 75 Pg 331 #18, 37

3 Trigonometry is A branch of geometry used first by the Egyptians and Babylonians (Iraq) Used extensively is astronomy and building Based on ratios between angles in RIGHT Triangles

4 The Trigonometric (trig) ratios: FUNCTION INVERSE FUNCTION

5 Also true are

6 Sample keystrokes Careful about Deg or Rad Setting Exit Ticket pg 369 #30-40 evens Sample keystroke sequences Sample calculator display Rounded Approximation sin sin 74 ENTER COS COS 74 ENTER 74 TAN TAN 74 ENTER

7 A swimmer sees the top of a lighthouse on the edge of shore at an 18º angle. The lighthouse is 150 feet high. What is the number of feet from the swimmer to the shore? 150 Tan 18 = = x 150 x x = X = ft x 18º

8 A dragon sits atop a castle 60 feet high. An archer stands 120 feet from the point on the ground directly below the dragon. At what angle does the archer need to aim his arrow to slay the dragon? Tan x = Tan x = 0.5 Tan -1 (0.5) = 26.6º 60 x 120

9 Ex. 5 A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river s edge at a 60 angle. How far must the person walk to reach the river s edge? cos x x (cos 60 ) = 200 x X = 400 yards Exit Ticket WS 2, 4, 10, 12, 18, 20 For 18 find values of all trig functions

10 An explorer is standing 14.3 miles from the base of Mount Everest below its highest peak. His angle of elevation to the peak is 21º. What is the number of feet from the base of Mount Everest to its peak? x Tan 21 = x 21º = 1 x = 5.49 miles = 29,000 feet x 14.3

11 The Trigonometric Functions SINE COSINE TANGENT

12 Greek Letter q Pronounced theta Represents an unknown angle

13 Greek Letter α Pronounced alpha Represents an unknown angle

14 Greek Letter β Pronounced Beta Represents an unknown angle

15 Finding Trig Ratios A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan respectively.

16 Trigonometric Ratios Let ABC be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows. A c b side adjacent to angle A B a C Side opposite angle A sin A = Side opposite A = a c Side adjacent to A cos A = = b c Side opposite A tan A = = a Side adjacent to A b

17 Opp Sin Hyp Adj Cos Hyp C q opposite B Tan Opp A adjacent Adj

18 We could ask for the trig functions of the angle by using the definitions. You MUST get them memorized. Here is a mnemonic to help you. b c The sacred Jedi word: SOHCAHTOA a adjacent opposite sin b c cos adjacent a c tan opposite adjacent b a

19 4 b It is important to note WHICH angle you are talking about when you find the value of the trig function. a 3 c 5 adjacent Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so a 2 Let's choose: b 2 c sin = o h 3 5 tan = o a 4 3 Use a mnemonic and figure out which sides of the triangle you need for sine. tangent.

20 You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The will always be the longest side and will always be opposite the right angle. 4 Oh, I'm acute! 5 This method only applies if you have a right triangle and is only for the acute angles (angles less than 90 ) in the triangle. 3 So am I!

21 Ex. 1: Finding Trig Ratios Large Small sin A = opposite cosa = adjacent tana = opposite adjacent B B A 7.5 C A 15 C Trig ratios are often expressed as decimal approximations.

22 Ex. 1: Finding Trig Ratios Large Small sin A = opposite cosa = adjacent tana = opposite adjacent B B A 7.5 C A 15 C Trig ratios are often expressed as decimal approximations.

23 Ex. 2: Finding Trig Ratios S sin S = opposite R coss = adjacent opposite 5 13 tans = opposite adjacent T 12 adjacent S

24 Ex. 2: Finding Trig Ratios S sin S = opposite R coss = adjacent opposite 5 13 tans = opposite adjacent T 12 adjacent S

25 Ex. 2: Finding Trig Ratios Find the sine, the cosine, and the tangent of the indicated angle. R sin S = opposite coss = adjacent tans = opposite adjacent R adjacent 5 13 T 12 opposite S

26 Ex. 2: Finding Trig Ratios Find the sine, the cosine, and the tangent of the indicated angle. R sin S = opposite coss = adjacent tans = opposite adjacent R adjacent 5 13 T 12 opposite S

27 Ex: 5 Using a Calculator You can use a calculator to approximate the sine, cosine, and the tangent of 74. Make sure that your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.

28 Tangent A = opposite adjacent 3.2 cm 7.2 cm 24º Tangent 24º 0.45

29 3.2 cm Sin α = 7.9 cm opposite 24º Sin 24º 0.41

30 Cosine β = adjacent 7.9 cm cm 46º Cos 46º 0.70

31 A plane takes off from an airport an an angle of 18º and a speed of 240 mph. Continuing at this speed and angle, what is the altitude of the plane after 1 minute? After 60 sec., at 240 mph, the plane has traveled 4 miles x 4 18º

32 SohCahToa Sine A = x opposite Sine 18 = opposite = 1 18º x 4 x 4 x = miles or 6,526 feet

33 Solving a Problem with the Tangent Ratio h =? We know the angle and the side adjacent to 60º. We want to know the opposite side. Use the tangent ratio: 60º 53 ft tan opp adj h 53 h 53 h 53 Why? 3 92 ft

34 Ex. A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as How tall is the tree?? tan 71.5 y tan y = 50 (tan 71.5 ) y = 50 ( ) y Opp Hyp ft

35 Notes: If you look back at Examples 1-5, you will notice that the sine or the cosine of an acute triangles is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the. The length of a leg or a right triangle is always less than the length of its, so the ratio of these lengths is always less than one.

36 Ex A 100 degree sector cut from a circular disc has a length of 7 cm. To the nearest cm., what is the radius of the circle? What is the area of the sector? S = r ɵ Must be in radians ɵ Just convert degrees to radians Ans. 4 cm Ans sq. cm

37 What is SohCahToa? Is it in a tree, is it in a car, is it in the sky or is it from the deep blue sea?

38 This is an example of a sentence using the word SohCahToa. I kicked a chair in the middle of the night and my first thought was I need to SohCahToa.

39 An example of an acronym for SohCahToa. Seven old horses Crawled a hill To our attic..

40 Old Hippie Some Old Hippie Came A Hoppin Through Our Apartment

41 SOHCAHTOA Sin Opp Hyp Cos Adj Hyp Old Hippie Tan Opp Adj

42 Other ways to remember SOH CAH TOA 1. Some Of Her Children Are Having Trouble Over Algebra. 2. Some Out-Houses Can Actually Have Totally Odorless Aromas. 3. She Offered Her Cat A Heaping Teaspoon Of Acid. 4. Soaring Over Haiti, Courageous Amelia Hit The Ocean And Tom's Old Aunt Sat On Her Chair And Hollered. -- (from Ann Azevedo)

43 Other ways to remember SOH CAH TOA 1. Stamp Out Homework Carefully, As Having Teachers Omit Assignments. 2. Some Old Horse Caught Another Horse Taking Oats Away. 3. Some Old Hippie Caught Another Hippie Tripping On Apples. 4. School! Oh How Can Anyone Have Trouble Over Academics.

44 Trigonometry Ratios Tangent A = opposite adjacent opposite Sine A = adjacent Cosine A = A Soh Cah Toa

45 14º 24º 60.5º 46º 82º

46 The Tangent of an angle is the ratio of the opposite side of a triangle to its adjacent side. 1.9 cm 1.9 opposite 7.7 cm adjacent 14º Tangent 14º 0.25

47 Tangent A = opposite adjacent 5.5 cm cm 46º Tangent 46º 1.04

48 Tangent A = opposite adjacent 6.7 cm Tangent 60.5º cm 60.5º

49 very large opposite Tangent A = adjacent As an acute angle of a triangle approaches 90º, its tangent becomes infinitely large Tan 89.9º = 573 Tan 89.99º = 5,730 etc. very small

50 Since the sine and cosine functions always have the as the denominator, and since the is the longest side, these two functions will always be less than 1. opposite Sine A = adjacent Cosine A = A Sine 89º =.9998 Sine 89.9º =

51 Ex. 3: Finding Trig Ratios Find the sine, the cosine, and the tangent of sin 45 = opposite 1 2 = cos 45 = adjacent 1 2 = tan 45 = opposite adjacent 1 1 = 1 Begin by sketching a triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. From Theorem 9.8 on page 551, it follows that the length of the is

52 Ex. 4: Finding Trig Ratios Find the sine, the cosine, and the tangent of sin 30 = opposite 1 2 = 0.5 cos 30 = adjacent tan 30 = opposite adjacent 1 3 = Begin by sketching a triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Theorem 9.9, on page 551, it follows that the length of the longer leg is 3 and the length of the is

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58 Trigonometric Functions on a Rectangular Coordinate System y Pick a point on the terminal ray and drop a perpendicular to the x-axis. r q x y x The adjacent side is x The opposite side is y The is labeled r This is called a REFERENCE TRIANGLE. sin q y r x cosq r y tan q x cscq r y r secq x x cotq y

59 Trigonometric Ratios may be found by: 2 45 º 1 1 Using ratios of special triangles sin cos tan 45 1 For angles other than 45º, 30º, 60º you will need to use a calculator. (Set it in Degree Mode for now.)

60 We need a way to remember all of these ratios