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Math 170 Trigonometry Lecture Notes Chapter 1 1.1 & 1.2 Angles, Degrees, Special Triangles, Distance Formula: Types of Angles: Acute Right Straight Obtuse Complementary Angles: Two angles that add to ____________ Supplementary Angles: Two angles that add to ____________ Find the supplement & complement (if they exist): a) 73⁰ b) 31⁰ c) 111⁰ d) x⁰ Sum of the Angles in a Triangle is: ___________ Use your knowledge of angles to solve for x and find the angle measures: 1) xº 2) (3x + 6)º (2x)º (1.5x + 10)º xº 1 Math 170 Trigonometry Lecture Notes Chapter 1 Angles in Trigonometry: • All angles start at the positive x – axis. o This is called: Standard Position o The initial side: o The terminal side: • Positive angles are counterclockwise. • Negative angles are clockwise. • Full circle is _____o. Straight line is ____o. Right angle is ____o . Coterminal Angles: • Any angle in standard position that ends at the same ray. • Differ by a multiple of __________. For any given ray, there are an infinite number of coterminal angles. a) Find 5 angles coterminal with 82⁰. b) Find 5 angles coterminal with -220⁰ Geometry Review Vertical Angles: (5x - 129)° (2x - 21)° Types of Triangles: based on Angles: Acute Right Obtuse based on Sides: Equilateral Isosceles Scalene 2 Math 170 Trigonometry Lecture Notes Chapter 1 Recall Similar Triangles: * Corresponding angles are ____________. * Corresponding sides are ______________. Given the following similar triangles, find the length of the missing sides: a) b) 10’ 14’ a b a 30" 20’ 60’ 50" 100" Use similar triangles to solve the following. Draw the picture first! a) A tree casts a shadow 30’ long. At the same time a 6’ tall man casts a shadow 4’ long. Find the height of the tree. b) A tower casts a shadow 150’ long. At the same time a 5’ tall woman casts a shadow 7.5’ long. Find the height of the tower. Pythagorean Theorem: Used to find third unknown side of right triangle Common Pythagorean Triples: 3: 5: 8: 7: 9: 4 12 15 24 40 :5 :13 :17 :25 :41 3 Math 170 Trigonometry Lecture Notes Practice a) Given one leg is 5cm and the hypotenuse is 20 cm Chapter 1 b) Given legs are 10 in and 24 in Distance Formula Really just a fancy form of _________________________ Practice: 1) Find the distance between (2, -3) and ( -1, -2) Special Right Triangles: 30o – 60o – 90o Isosceles Right Triangle Example: If the long leg of 30-60-90 triangle is 8cm, find the length of the other sides. Rational Expression Review: 1) Simplify 252 = 2) Rationalize the denominator 12 10 CA 1.2 pg 25 #69, 73; 1.1 pg 12-13 #27, 41, 53, 67; 3) Solve for x 3 x=7 Special Δ worksheet 4 Math 170 Trigonometry Lecture Notes Chapter 1 Similar Triangles are the basis for trigonometric functions. We know that if we have similar triangles we always have the same ratio of the sides. These ratios are actually given by trigonometric functions so you won’t have to have the “other triangle” to compare to: the trig function gives you the ratio of the similar triangle. However, to use them we need (1) a right triangle and (2) the measure of one more angle in the triangle 1.3 Trigonometric Functions & 1.2 Coordinate System SOH CAH TOA Right Triangles Angles (in Standard Position) y P (x, y) θ x Function Right Triangles Angles Function sin θ csc θ cos θ sec θ tan θ cot θ Right Triangles Angles Find the 6 trigonometric functions of ϴ, for the following points on the terminal side of ϴ: • Draw the triangle. • Find r (use Pythagorean Thm) a) P (-3, 4) b) P (-5, - 12) 5 Math 170 Trigonometry Lecture Notes Chapter 1 Signs in each Quadrant: All Students Take Calculus Function Values of Quadrantal Angles: θ 0⁰/ 360⁰ 90⁰ 180⁰ 270⁰ sin θ csc θ cos θ sec θ tan θ cot θ Find the following: (No calculator) a) 4 csc 270⁰ + 3 cos 180⁰ b) sin 180⁰ + cos2 180⁰ Slopes, lines, parabolas Review on your own: pg 15-16 Graphing lines: y = mx + b 6 Math 170 Trigonometry Lecture Notes Chapter 1 Using Trig Functions Find the remaining trig functions, given one trig fn and the quadrant. • Sketch an angle in the appropriate quadrant. • Draw a triangle and use the given trig fn to find two sides of the triangle. • Use Pythagorean Thm. to find the other side. a) cos θ = 5 θ is in Quad. IV 8 b) csc θ = 2 θ is in Quad. II c) sin θ = 2 6 d) tan θ = −⅔ sin θ > 0 cos θ < 0 Range of Trigonometric Fns: (pg 29) • Sin and Cos are between ________ and _________ • Csc and Sec are less than ________ or greater than ___________ • Tan and Cot can be any value. Is it possible? a) sin θ = −¾ b) cos θ = 2.5 c) tan θ = 150 d) csc θ = −0.5 CA: 1.3 pg 32 #45, 59, 65, 69; 1.2 pg 25 #21, 48 1.4: 15, 19, 23, 33, 41 1.5: 5, 9, 19, 31, 37, 51, 59, 67, 75, 93 7 Math 170 Trigonometry Lecture Notes Chapter 1 HW Page: 1.4 Introduction to Trig Identities Find these identities in your book and write them here. Reciprocal Identities: (pg 34) Ratio Identities: csc ϴ = sin ϴ = tan θ sec ϴ = cos ϴ = cot θ tan ϴ = cot ϴ = (pg 35) Pythagorean Identities Watch videos from course webpage: http://www.elcamino.edu/faculty/ammartinez/Math170.html • Trig Identities A L2 (4:49) • Trig Identities B L3 (4:57) Take notes on the proofs and then write the identities in the box. (Do NOT just write the identities.) Notes: The Three Pythagorean Identities CA: Do the following using identities (NOT Pythagorean Theorem): 1.4 15, 19, 23, 33, 41 1.5 5, 9, 19, 31, 37, 51, 59, 67, 75, 93 8