Warm Up: please factor completely

Transcription

1 Warm Up: please factor completely

2 vocabulary KEY STANDARDS ADDRESSED: MA3A2. Students will use the circle to define the trigonometric functions. a. Define and understand angles measured in degrees and radians, including but not limited to 0, 30, 45, 60, 90, their multiples, and equivalences. b. Understand and apply the six trigonometric functions as functions of general angles in standard position. c. Find values of trigonometric functions using points on the terminal sides of angles in the standard position. d. Understand and apply the six trigonometric functions as functions of arc length on the unit circle. e. Find values of trigonometric functions using the unit circle.

3 Standard: Understand and apply the six trigonometric functions as functions of general angles in standard position. 5.1 Angles and Degree Measure Sketch an angle and lable the following: vertex, initial side, terminal side Standard position: Degree: Minute: Second: 1 degree = 1/360 of a circle..in other words a full circle is 360 degrees 1 degree = 60 minutes 1 minute = 60 seconds

4 Standard: Understand and apply the six trigonometric functions as functions of general angles in standard position. How do you convert into DMS (degree minutesecond form)? Converting Degrees, Minutes, & Seconds to Degrees & Decimals To convert degrees, minutes, and seconds (DMS) to degrees and decimals of a degree (DD): 1. First: Convert the seconds to a fraction. Since there are 60 seconds in each minute, 37 42' 17" can be expressed as /60'. Convert to '. 2. Second: Convert the minutes to a fraction. Since there are 60 minutes in each degree, ' can be expressed as /60. Convert to

5 Standard: Understand and apply the six trigonometric functions as functions of general angles in standard position. Convert these DMS to the DD form. Round off to four decimal places Questions Answers (1) 89 11' 15" (2) 12 15' 0" (3) 33 30' 33.5 (4) 71 0' 30" (5) 42 24' 53" (6) 38 42' 25" (7) 29 30' 30" (8) 0 49' 49"

6 Standard: Understand and apply the six trigonometric functions as functions of general angles in standard position. Converting Degrees & Decimals to Degrees, Minutes, & Seconds To convert degrees and decimals of degrees (DD) to degrees, minutes, and seconds (DMS), referse the previous process. 1. First: Subtract the whole degrees. Convert the fraction to minutes. Multiply the decimal of a degree by 60 (the number of minutes in a degree). The whole number of the answer is the whole minutes. 2. Second: Subtract the whole minutes from the answer. 3. Third: Convert the decimal number remaining (from minutes) to seconds. Multiply the decimal by 60 (the number of seconds in a minute). The whole number of the answer is the whole seconds. 4. Fourth: If there is a decimal remaining, write that down as the decimal of a second. Example: Convert to DMS = is the whole degrees x 60' per degree = ' 14 is the whole minutes ' x 60" per minutes = 4.416" 4.416" is the seconds DMS is stated as 5 14' 4.416"

7 Standard: Understand and apply the six trigonometric functions as functions of general angles in standard position. Convert these DD to the DMS form. Questions Answers (1) ' 0" (2) ' 30" (3) ' 45" (4) ' 52.2" (5) ' 24.42" (6) ' 0" (7) ' 21.12" (8) ' 0.96"

8 Standard: Understand and apply the six trigonometric functions as functions of general angles in standard position. 1.Convert degrees to DMS form 329 ο 7' 30'' 2. Convert to decimal form Name all four basic quandrantal angles. 90, 180, 270, 360 (or 0) 4. Give the angle measure represented by each rotation: a) 3.5 rotations clockwise 3.5 * ( 360) = 1260 Clockwise rotations have negative measure. The angle measure of 3.5 clockwise rotations is b) 4.25 rotations counterclockwise 4.25 * 360 = 1530 Counterclockwise rotations have positive measure. The angle measure of 4.25 counterclockwise rotations is 1530.

9 Coterminal Angles: Standard: Find values of trigonometric functions using points on the terminal sides of angles in the standard position. are angles in standard position (angles with the initial side on the positive x axis) that have a common terminal side. For example 30, 330 and 390 are all coterminal. To find a positive and a negative angle coterminal with a given angle, you can add and subtract 360 if the angle is measured in degrees. Example 1: Find a positive and a negative angle coterminal with a 55 angle = = 415 A 305 angle and a 415 angle are coterminal with a 55 angle. Give 2 angles that are coterminal to 45 degrees = = 315

10 Reference angle: Standard: Find values of trigonometric functions using points on the terminal sides of angles in the standard position. The reference angle is the acute angle formed by the terminal side of the given angle and the x axis. Reference angles may appear in all four quadrants. Angles in quadrant I are their own reference angles. Remember: The reference angle is measured from the terminal side of the original angle "to" the x axis (not the y axis). give the reference angle for the following: a) 102 degrees = 78 b) 320 degrees = 40 c) 125 degrees first convert to a positive angle = 235 then find the ref angle = 55

11 practice Standard: Find values of trigonometric functions using points on the terminal sides of angles in the standard position Find a positive coterminal angle smaller than 360 o to angles a) A = 700 o = = 20 b) B = 940 o = = 220

12 Standard: Find values of trigonometric functions using points on the terminal sides of angles in the standard position. right triangle trig review: What do you remember

13 Standard: Find values of trigonometric functions using points on the terminal sides of angles in the standard position. 5.2 Trigonometric Ratios in Right Triangles SOH CAH TOA Do they always work? No! The basic trig rules only apply to RIGHT triangles click the link to take notes on SOH CAH TOA

14 Standard: Find values of trigonometric functions using points on the terminal sides of angles in the standard position. The six basic Trig functions

15 C 8 cm 17 8/17 Sin A = Cos A = 15/17 Tan A = 8/15 B 15 cm A Csc A = 17/8 first use pythagorean theorem to find the missing side = c 2 so side c = 17 Sec A = 17/15 Cot A = 15/8

16 y Solve using trig to find the values of x and y To find x: 1. what trig function do you need? (always go from the given angle)

17 Warmup Given the following locations convert the latitude and longitudes another format. Model High School is 34 o 18' 25" N 85 o 05' 43" W Armuchee High School is o N o W o N o W 34 o 20' 35" N 85 o 10' 21" W Coosa High School is 34 o 15' 48" N 85 o 18' 15" W o N o W

18 If sec θ = 3.05, find cos θ. If tan θ = 2/5, find cot θ.

19 y Solve using trig to find the values of y and x

20 Let's Practice!

21 Do you remember your special right triangles from geometry? click for review and rules If SL = 5, find LL and H LL = SL * 3 so LL = 5 3 H = SL * 2 so H = 5*2 = 10 find x: click for review and rules Pattern Formula solution We are looking for the hypotenuse so we will use the pattern formula that will give the answer for the hypotenuse: Substituting the leg = 7, we arrive at the answer: 7 2 A nice feature of the pattern formulas is that the answer is already in reduced form.

22 5.1 and 5.2 answers

23 Warm UP: factor completely:

24 5.3 The unit circle and triangle trigonometry Standard: Defines the six trig functions as both circular functions and ratio of sides in right triangles EQ: What is the Unit circle? How can triangles help us define values on the unit circle? Unit Circle Power Point!

25 The Unit Circle 5.3 What is the "Unit Circle" If the terminal side of an angle in standard position intersects the Unit Circle at the point P(x,y) then the cosine of the angle is and the sine of the angle is Using the unit circle with (x,y) the six basic trig functions can be defined as: Cos(θ) = x Sec (θ) = 1/x Sin(θ) = y Csc (θ) = 1/y Tan (θ) = y/x Cot (θ) = x/y

26 Review of Special Right Triangles 45º 45º 90º Triangles 30º 60º 90º Triangles Right Triangles go to attachments to get hyper link for power point

27 Special right triangles and the unit circle Derive the values for sin, cos, and tan for 30, 45, 60, 90, 180, 270, and 360 using the special right triangles final product

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29 Standard : Defines the six trig functions as both circular functions and ratio of sides in right triangles For any angle in standard position with measure n degrees and the point (x,y) on its terminal side where r= x 2 + y 2 sin n = y/r csc n = r/y cos n = x/r tan n = y/x sec n = r/x cot n = x/y

30 Standard : Defines the six trig functions as both circular functions and ratio of sides in right triangles Find the six trig values of an angle in standard position if a point with the coordinates ( 15,20) lies on its terminal side (draw a triangle)

31 Standard : Defines the six trig functions as both circular functions and ratio of sides in right triangles Suppose angle n is in standard position and its terminal side is in quadrant 4. If sec n = 10/3 find the values of the five other trig functions

32 Standard : Defines the six trig functions as both circular functions and ratio of sides in right triangles Unit Circle links unit circle: unit circle practice: converting deg rad/ defn of radian: unit circle all kinds of thing:

33 5.3 answers

34 1. Warm UP! Use the unit circle to find each value. 2. a. tan 180 b. csc ( 90 ) Use the unit circle to find the values of the six trigonometric functions for a 300 angle. 3. Find the values of the six trigonometric functions for angle θ in standard position if a point with coordinates ( 6, 8) lies on its terminal side. cos x = 6/10 or 3/5 sec x = 5/3 sin x = 8/10 or 4/5 csc x = 5/4 tan x = 8/6 or 4/3 cot x = 3/4

35 Standard : Defines the six trig functions as both circular functions and ratio of sides in right triangles 5.4 Right Triangle Trig Use SOH CAH TOA to solve each problem

36 Standard : Defines the six trig functions as both circular functions and ratio of sides in right triangles The angle of elevation is always measured from the ground up. Think of it like an elevator that only goes up. It is always INSIDE the triangle. The angle of depression is always OUTSIDE the triangle. It is never inside the triangle. But you use the inside angle to to complete the triangle!

37 Standard : Defines the six trig functions as both circular functions and ratio of sides in right triangles Example 1 If B = 42 and a = 12, find c. c is about 16.1

38 Standard : Defines the six trig functions as both circular functions and ratio of sides in right triangles Example 2 RECREATION A child holding on to the string of a kite gets tired and decides to put the string on the ground and secure it with a brick. The length of the string from the brick to the kite is 240 feet. a. If the angle formed by the string and the ground is , how high is the kite? The kite is about feet high b. What is the horizontal distance between the kite and the brick? feet

39 Standard : Defines the six trig functions as both circular functions and ratio of sides in right triangles Example 3 GEOMETRY A regular hexagon is inscribed in a circle with diameter centimeters. Find the apothem of the hexagon. The apothem is about 4.47 centimeters.

40 Warm up for sec 5.5

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43 5.5 Solving Right Triangles *If you know the sides and you are looking for the angle in a triangle, you use the unit circle or inverse trig on the calculator. sin 1 x, cos 1 x or tan 1 x on the calculator are the inverse buttons a B C b c Solve each problem based on the given information: 1. If A=38 and a = 14, find c 2. If B = 58 and c = 32, find a. A 3. If b = 45 and B = 72, find a. 4. If A = ' and c = 27, find b

44 A regular hexagon has side of 8 cm. Find the length of the apothem to the nearest tenth. Draw a picture.

45 Solve each equation: Think in terms of the unit circle! 1. tan x = 1 4. Name 4 angles whose cosine = 1/2 2. sin x = 1/2 5. Cos x = 0 3. cos x = 3 / 2 Evaluate each expression: 1. cos(cos 1 2/3) = 3. csc( arcsin 1/2) = 2. tan(cos 1 5/13)=

46 Draw a diagram and solve each triangle 1. C = 90, A = 40, a = C=90, c=65, b=55

47 1. If a 100 ft. building casts a 88 ft. shadow, what is the angle of elevation of the sun A security light is being installed outside Mrs. Staud's house. The light is mounted 20 ft. above the ground and she wants to be sure that is will illuminate the end of her driveway, which is 100 ft. long. What should be the angle of depression of the light? 11.3

48 Practice!

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51 Fill in the unit circle Label the degree measure, x value, and y value for each mark name

52 A Warmup! b C a c B A = b = a = Given that c=15.2 and B = 33 degrees solve the triangle

53 Law of Sines Lesson LS is useful when you do not have a right triangle! It can be used to solve any type of triangle as long as you have (or can get) a "matching pair". law_of_sines.ppt

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56 In, m<r = 105º, r = 12, and t = 10. Find the m<s, to nearest degree.

57 In, m<b = 80º, m<c = 34º and a = 16. Find the length of b to the nearest tenth.

58 In, sin P = 0.2, sin R = 0.4 and r = 22. Find the length of p.

59 Ambiguous Case Law of Sines 1. You only have to worry about two possible triangles if you are given one acute angle 2. If you are given two angles, there will always be one triangle formed! 3. If you are given one angle (either Obtuse or Acute) it is possible that no triangle can be formed! Watch out for "ERROR" on the calculator!

60 ASA AAS SSA Obviously you will need to know bsina (the height of the triangle). if a is the smallest there is This is called the Ambiguous Case not a triangle. because there are two distinct correct solutions (side a is larger than the height). This is because the largest side must be across from the largest angle.

61 For SSA Setup If angle θ is acute: adj θ opp If opp side is greater than the adj only 1 solution If opp side is less than adj side: run the test value for triangle height h = adj(sin θ) If opp side = h If opp side < h If opp side > h 1 solution and Δ is right adj θ h opp No solution side too short adj θ h opp 2 Δ's formed adj h θ opp

62 Law of Sines Examples for the ambiguous case

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64 In triangle ABC, a = 20, c = 16, and m<a = 30º. Solve the triangle: B A c

65 In triangle ABC, a = 7, c = 16, and m<a = 30º, solve the triangle:

66 In triangle ABC, a = 10, b = 16, and m<a = 30º Solve the triangle: A 30 B c

67 click for lesson Find the area of a triangle with trigonometry!

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72 Attachments Acc. Math III unit 4 SE.pdf 5.2 notes.notebook Unit_Circle.pdf Unit_Circle_.ppt AM3 factoring quiz show me some work!.ia2 law_of_sines.ppt