Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Math 170 Trigonometry Lecture Notes Chapter 2 2.1 Trigonometric Functions of Acute Angles Note in the right triangle below, the two non-right angles are always going to be acute. The basis of our trig functions work with these acute angles. We will deal with non-acute angles separately. Cofunctions Sine <A = Cosine < B = A c Cosecant <A = Secant < B = Tangent <A = Cotangent <B = b Cofunction Theorem: A trigonometric function of an angles is always ______________ the cofunction of the ___________________ of the angle. C a B In General: For any angle, θ Sin θ = Cos (90o – ___) Csc θ = Sec ( Write each in terms of its Cofunctions: 1) Sec 73⁰ = 2) Cot 18⁰ = Use Cofunctions to Solve for θ 4) Sin (2θ + 10) = Cos (3θ – 20) Function ) Tan θ = Cot ( ) 3) Cos 26⁰ = 5) Sec (θ – 20) = Csc (θ + 40) Cofunction Functions of Special Triangles: 1) 30-60-90 Start with an equilateral triangle: Sin 30⁰ = Csc 30⁰ = Cos 30⁰ = Sec 30⁰ = Tan 30⁰ = Cot 30⁰ = 1 Math 170 2) 45-45-90 Trigonometry Lecture Notes Chapter 2 Start with a square: Sin 45⁰ = Cos 45⁰ = Tan 45⁰ = Csc 45⁰ = Sec 45⁰ = Cot 45⁰ = CA: 2.1 15, 41, 43, 53 2.2 Finding Trig Values using a Calculator Rounding warning: In the middle of calculations you should leave ALL decimal places. Do NOT round on intermediate steps, only at the end. ***Pay attention to this. It applies ALL semester. Don’t lose exam and quiz points by ignoring this!!!!! *** Degrees, Minutes, & Seconds (DMS): Degree: 1⁰ contains: ________ minutes Minute: 1’ contains: ________ seconds 1⁰ contains: __________ seconds Adding with DMS: • Remember the maximum value for minutes & seconds is __________. • If something adds to 67’ that is ______ degrees and ______ minutes. a) 25⁰ 37’ 19’’ + 35⁰ 51’ 42” b) 11⁰ 24’ 49’’ + 16⁰ 39’ 50” Subtracting with DMS: Borrowing: • To borrow minutes: Make the degree _____ less, but add _______ minutes. • To borrow seconds: Make the minutes _____ less, but add ______ seconds. a) 73⁰ 21’ 18’’ – 45⁰ 48’ 59” b) 97⁰ 11’ 8’’ – 48⁰ 19’ 57” 2 Math 170 Trigonometry Lecture Notes Converting from DMS to Decimals: Just Add: Degree + a) 73⁰ 21’ 18’’ Chapter 2 Minutes Seconds + 60 3600 b) 48⁰ 19’ 57” Converting from Decimals to DMS A. Leave the whole number (degree) alone. B. Multiply the decimal part by 60, this gives you the minutes. C. Look only at the decimal part of the minutes. a. Multiply by 60, this gives you the seconds. b. Round to the nearest whole number. a) 85.1234⁰ b) 11.9876 Calculator Use - Things to Remember: 1) Change DMS to decimals 2) Only sin, cos, & tan are on calculator. Csc θ = _________ Sec θ = __________ Cot θ = ___________ Change to sin, cos, or tan before using trigonometric functions. 3) Make sure the Mode is “Degree” 4) Use Order of Operations Use the Calculator to Find: 1) cot 41˚ 6’ 2) sec 13˚ 15’ Finding Inverse Trig Functions: sin-1 says “What angle gives us this sin” 1) tan θ = 2.345 2) csc θ = 4.56 3) 1 sec 15° 3) sec θ = -2.1 CA 2.2 35, 47, 65 3 Math 170 Trigonometry Lecture Notes Chapter 2 2.3 Solving Right Triangles We can find all the sides and angles of a right triangle, if we are given: a) Two sides b) One side, and one of the acute angles Let <C be a right angle. Find the missing sides & angles: (Round SOLUTIONS to two decimal places.) 1) <A = 25⁰ 10’ and b = 20’ ft A c b C 2) a = 12.5 cm and c = 18.9 cm a B A c b C 3) <B = 49⁰ 25’ and c = 3.25 m a B A c b C a B 4) The base of an isosceles triangle is 10 m and the two congruent angles are 40⁰. Find the height of the triangle and the lengths of the congruent sides. DRAW the pic first! CA 2.3: #13, 17, 27, 37, 40, 41, 58 4 Math 170 Trigonometry Lecture Notes Chapter 2 2.4 Applications Example: 2.4 #15 Angles of Elevations and depression: (Video: Definition of Angle Depression (2:14) - silent video) CA: Video: Angles of Elevation and depression (6:29) Follow along by taking notes in your CA book. 1) The angle of elevation from the top of a small building to the top of a nearby taller building is 46⁰ 40’, while the angle of depression to the bottom is 14⁰ 10’. If the shorter building is 28 m high, find the height of the taller building. CA 2.4 # 58, 60 Bearings: Used in navigation Video: ________________________________ Two ways to tell bearings: a) Degrees clockwise from N b) Tell N or S first, then the degrees E or W Describe the Bearings of the following points using both methods: 1) (-3, 0) 2) (-7, -7) 3) (-4, 4) 5 Math 170 Trigonometry Lecture Notes Chapter 2 Use Bearings to Solve Problems: 1) A ship travels at 55 km/hr for 2 hours at a bearing of 27⁰. Then it turns and travels 60 km/hr for 1.5 hours at a bearing of 117⁰. How is the ship from its starting position? 2) Two ships leave port at the same time. One travels 24 mph at a bearing of S 61⁰ 51’ E. The other travels 28 mph at a bearing of N 28⁰ 10’ E. How far apart are the ships after 4 hours? CA Problems: Other Applications: Finding Heights or Distance 1) On a trip to Mt. Rushmore, you’d like to find out the length of Pres. Washington’s face. Standing, 175 feet from the mountain, the angle of elevation to the top of his head is 45.8⁰. The angle of elevation to the bottom of his chin is 34.4⁰. How long is his face? CA 2.4 # 27, 29, 57, 59 Geometry Review: Parallel lines cut by a transversal: t 1 m 3 n 5 7 2 4 6 8 6 Math 170 Trigonometry Lecture Notes Chapter 2 2.5 Vectors Video Clip: https://www.youtube.com/watch?v=bOIe0DIMbI8 Vectors quantities are quantities that have both ____________________ and ___________________. Notation V V x |V| Vectors are equivalent if they have the same __________________ and _____________________ Adding & Subtracting Vectors. Place the vectors _________ to tail. Complete the ___________________. This resultant vector is the “sum of the vectors” Find the sum of these: For subtraction, you reverse the direction of the vector that is negated, then add. Example 1: A boat is crossing a rive that runs due north. The boat is pointed due east and is moving through water at 12 mph. If the current is a constant rate of 5.1 mph, find the actual course of the boat through the water. Round solution to two decimal places. 7 Math 170 Trigonometry Lecture Notes Chapter 2 Horizontal & Vertical Vector Components: Place the vector in standard position. Given |V| |V x |= Given |V x | or |V y | |V| = and, |V y |= Magnitude of a vector can also be given by Pythagorean Theorem / distance formula : |V| = |V| = CA: 2.5 #21, 27, Practice together: 2.4 #30 2.5 #32 CA 2.5 #17, 34 8