Chapter 2. Right Triangles and Static Trigonometry

Transcription

1 Chapter 2 Right Triangles and Static Trigonometry 1

2 Chapter 2.1 A Right Triangle View of Trigonometry 2

3 Overview Values of six trig functions from their ratio definitions Bridge definitions of trig functions by placing a triangle in the coordinate plane Use cofunctions to write equivalent expressions 3

4 An Alternate Terminology In defining the trig functions, we used the coordinate plane that wasn t necessary For a given triangle, we can define for each angle: Opposite side (opposite angle) Adjacent side (side of the angle) Hypotenuse (long side of triangle) The triangle can now be in any orientation not just along an axis Hyp Opp Adj 4

5 Re-Definition of Trig Functions Let: Opp = Opposite Adj = Adjacent Hyp = Hypotenuse Then: sin = opp/hyp cos = adj/hyp tan = opp/adj csc = hyp/opp sec = hyp/adj cot = adj/opp SOHCAHTOA 5

6 Example For a triangle opp hyp In this triangle, = 45 adj Let the adj side = x; then opp = x, hyp = x 2 sin = 1/ 2 cos = 1/ 2 tan = 1 6

7 A Triangle 60 a c 90 b 30 Suppose side a = 1; we have c = 2, b = 3 sin 30 = a/c =1/2 cos 30 = b/c = 3/ 2 tan 30 = a/b = 1/ 3 csc 30 =? sec 30 =? 7 cot 30 =?

8 A Triangle 60 a c 90 b 30 Suppose side a = 1; we have c = 2, b = 3 sin 30 = a/c =1/2 cos 30 = b/c = 3/ 2 tan 30 = a/b = 1/ 3 csc 30 = 2/ 3 sec 30 = 2 cot 30 = 3 8

9 A Triangle 60 a c 90 b 30 Suppose side a = 1; we can find c = 2, b = 3 sin 60 = 3/ 2 cos 60 = 1/2 tan 60 = 9

10 A Triangle 60 a c 90 b 30 Suppose side a = 1; we have c = 2, b = sqrt (3) sin 60 = 3/2 cos 60 = 1/2 tan 60 = 3 10

11 How to remember if sin 30 or cos 30 = 1/2 Triangle with angles A, B, C and sides a, b, c B c a C b A is opposite a, etc. A C is a right angle, c is the hypotenuse Sin A = a/c, sin B=b/c, etc. 11

12 Continued If a is larger than b, sin a is larger than sin b, etc. Which is larger: ½ or 3/2? Since 3 is larger than 1, 3/2 is larger than ½ One is the sine of 60, one of 30. Which is which? The sin 30 is ½, the smaller value since 30 is smaller than 60 For cosine, it is opposite. Cos 60 is ½ 12

13 Example Suppose sin = 2/3 What are cos and tan? 13

14 Solution Suppose sin = 2/3 What are cos and tan? sin = opp/hyp if opp = 2 and hyp = 3, what is adj? a 2 + b 2 = c 2 ; 4 + x 2 = 9 adj = sqrt (5) sin = 2/3 cos = sqrt (5)/3 tan = 2/sqrt (5) 14

15 Example Suppose cos = 5/13; find sin and tan 15

16 Solution Suppose cos = 5/13; find sin and tan sin = 12/13 tan = 12/5 16

17 Right Triangles and the Coordinate Plane We can always rotate a triangle to be in the coordinate plane 1. Find the correct ratio of the legs (the same as tan ) 2. Place the triangle in standard position, that is with at the origin and the right angle to the right 17

18 Example The two known sides are 3 and 5 What is tan?

19 Solution The two known sides are 3 and 5 What is tan? = 3/

20 Angle Complements What is the sum of the angles in a triangle? If we have a right triangle, how many degrees can be shared by the two non-right angles? 20

21 Angle Complements What is the sum of the angles in a triangle? 180 If we have a right triangle, how many degrees can be shared by the two non-right angles? 90 21

22 Angle Complements, Con t c a This means that if the angles of a triangle are,, and 90, = 90, = 90 sin = b/c, cos = b/c, so sin = cos The sine of an angle equals the cosine of its complement b 22

23 Cofunctions Cos and sin are called co-functions Tan and cot are also cofunctions The value of a function is the value of the complement of its cofunction sin (90-x) = cosx 23

24 Another Identity Verify cos 2 = 1 - cos 2 if and are compliments 24

25 Another Identity, Solution Verify cos 2 = 1 - cos 2 if and are compliments If and are compliments, then cos = sin Substituting, we have: cos 2 = 1- sin 2, which is the same as 1 = cos 2 + sin 2, the Pythagorean identity Therefore, the identity is true 25

26 Using Complements Find x so that sin 12 = cos x csc 17 = csc x cos 2x = sin 3x csc(6x 3) = sec(2x + 5) 26

27 Using Complements Find x so that sin 12 = cos x; x=78 csc 17 = sec x; x=73 cos 2x = sin 3x; x=18 csc(6x 3) = sec(2x + 5); x=11 27

28 Summary Can express trig functions in terms of the lengths of opposite, adjacent, and hypotenuse Can position a triangle in the coordinate plane Can use cofuctions and complements 28

29 Chapter 2.2 Solving Right Triangles 29

30 Overview Solve a right triangle, given one angle and one side Solve a right triangle given two sides 30

31 What is Solving a Triangle? To solve a triangle, we need: The measure of each of the three angles The measure of each of the three sides The convention we will use is Capital letters or Greek letters are for angles Lowercase letters are for sides To solve triangles, we will use The Pythagorean theorem A calculator, for trig functions Complements and Identities 31

32 Example Given: = 30; a = 17.9 c We know sin 30 = 17.9 / c, or c = 17.9/sin 30. a b sin 30 = 1/2, so c = 35.8 b = 17.9 sqrt(3) 31 32

33 Finding Values of Trig Functions Things to watch out for: The sign of the function. You should determine the correct quadrant before you put the numbers into your calculator Degrees vs. radians 33

34 Degrees vs. Radians What is a radian? The 360 degree circle is the same as 2 radians X degrees = 2 (X/360) radians How can you check how your calculator is set? Evaluate a number you know, like Sin 90 (should = 1) There should be a way on your calculator to switch Unless otherwise indicated, assume all angles are given in degrees in this class 34

35 Degrees to Radians X degrees = 2 (X/360) radians 90 deg = 2 (90/360) = 2 (1/4) = /2 radians 180 deg = 2 (180/360) = 2 (1/2) = radians 60 deg =? 45 deg =? 35

36 Solution X degrees = 2 (X/360) radians 90 deg = 2 (90/360) = 2 (1/4) = /2 radians 180 deg = 2 (180/360) = 2 (1/2) = radians 60 deg = /3 radians 45 deg = /4 radians 36

37 Given: = 58, a = 24 If = 58, then = 32 Example a Tan = tan 32 = 24/b, or 24 = b tan 32 b c What is tan 32? Use a calculator, tables, excel, etc. tan , so b 38.4 For side c: sin 32 = 24/c sin , so c

38 Example = 49, b = 89 a b c 38

39 Example (Solution) = 49, a = 89 + = 90, so = 41 To find a: tan = b/a a b c tan = 0.87, so b = a tan 41 = 77.4 To find c a 2 + b 2 = c 2 c = sqrt( ) = a = 89, b = 77.4, c =

40 Solving Right Triangles Given Two Sides Given two sides, we can always find the third: a 2 + b 2 = c 2 Usually need a calculator or tables to find the angles a c b Example: a=17, b=25 so tan = 17/25 How do we find? We need the inverse tan, notated arctan(a/b) or tan -1 (a/b) the arc whose tangent is Note: tan -1 is not 1/tan!!! It is the inverse of tan 40

41 Solving Right Triangles Given Two Sides, Cont. Given two sides, we can always find the third: a 2 + b 2 = c 2 a=17, b=25 so tan = 17/25 a b c arctan (17/25) 34.2 [tan /25 check ] since = 34.2, = = 55.8 c = sqrt[ (17) 2 + (25) 2 ]

42 a =221, b = 207; find angle, c Example a b c 42

43 a = 221, b = 207; find angle Tan = a/b = 221/207 = 1.07 arctan 1.07 = 46.9 Example (Solution) a c = sqrt (a 2 + b 2 ) = sqrt ( ) = b c 43

44 Summary Can solve a triangle given an angle and a side Can solve a triangle given two sides Side subjects (for later): Radians Inverse trig functions 44

45 Chapter 2.3 Applications of Static Trigonometry 45

46 Overview Solve applications involving elevation and depression Solve applications involving angles of rotation Solve general applications of right triangles 46

47 Terms Elevation something up Depression something down Where you are to where you look Both need an line of orientation, where you are Line of orientation Line of orientation Here is the angle of elevation Here is the angle of depression 47

48 Example You pitch your tent 250 yds from the base of a cliff. If the angle of elevation from your tent to the top of the cliff is 40 deg, how high is the cliff? 48

49 Example (Solution) You pitch your tent 250 yds from the base of a cliff. If the angle of elevation from your tent to the top of the cliff is 40 deg, how high is the cliff? 40 deg H =? 250 yds H/250 = tan (40) = 0.84; H = 0.84(250) = yds 49

50 Example From the top of a cliff 30 m high, you see a boat on a lake. The angle of depression is 35 deg. What is the distance from the cliff to the boat? 50

51 Example (Solution) From the top of a cliff 30 m high, you see a boat on a lake. The angle of depression is 35 deg. What is the distance from the cliff to the boat? 30m 35 deg Distance, d BOAT = = 55 deg Tan = d / 30 d = 30 tan 55 = 30 (1.4) = 42.8 m 51

52 Angles of Rotation The text give the example of angles w.r.t. north and south The key is that we can have an angle in any orientation and can move the triangle to standard position 52

53 Rotation Example The angle is 60 deg from north North 60 We can rotate the angle to standard position 60 53

54 Example A State Trooper hides 50 ft from the roadway in order to catch speeders. The angle between her and a road sign in the distance is 79 deg. She uses a stop watch to determine how long it takes a vehicle to pass her position and reach the road sign. She clocks 3 vehicles: find their speed. An 18 wheeler, 2.7 sec A truck, 2.3 sec Road Sign A car 1.9 sec 50 ft 79 deg Trooper 54

55 Solution Speed = Distance/Time = x/time x Road Sign x/50 = tan(79) = 5.14 x = ft 79 deg Trooper Now, the hard part we have feet and seconds and need mph! (1 ft/sec) (1 mile /5280ft ) (3600 sec/ 1 hour) = 0.68 m/hr 18 Wheeler speed: 257ft/2.7sec = 95.2 ft/sec = 65 mph Truck speed: 257 ft/2.3 sec = 76 mph Car speed: 257 ft/1.9 sec = 92 mph 55

56 More Examples In 2001 the tallest building was the Petronas Tower in Kuala Lumpur. For someone standing 25.9 ft from the base, the angle of elevation to the top of the tower is 89 deg. How tall is the tower? 56

57 Solution In 2001 the tallest building was the Petronas Tower in Kuala Lumpur. For someone standing 25.9 ft from the base, the angle of elevation to the top of the tower is 89 deg. How tall is the tower? x/25.9 = tan (89) x = ft x 89 deg

58 Example Using the same Petronas Tower, someone on the top sees her residence at 5 degrees depression. How far away is the residence from the base of the tower? Give answer in feet and miles. 58

59 Solution Using the same Petronas Tower, someone on the top sees her residence at 5 degrees depression. How far away is the residence from the base of the tower? Give answer in feet and miles. 5 deg ft 85 deg Distance to residence Tan(85) = x/ x = 16,900 ft 1690 ft (1 mile/5280ft) = 3.2 mi 59

60 Summary Need to know angle of elevation and depression From there the solution is just solving triangles Make pictures Watch units!!!!! 60

61 Chapter 2.4 Summary 61

62 Finding Reference Angles Using reference angles Solving applications Overview 62

63 References Angles The angle formed by the terminal side of an angle and the nearest x axis is the reference angle. A reference angle is always between 0 and 90 To calculate: In quadrant 1? In quadrant 2? In quadrant 3? In quadrant 4? 63

64 Examples Find the reference angles for the given angles:

65 Solutions Find the reference angles for the given angles:

66 Find the reference angles More Examples

67 Find Quadrant To Find Values of Trig Functions Find reference Angle 67