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Chapter 6.1 – 6.3 An Introduction to Trigonometry Page 36 1 Note • This presentation contains material not explicitly given in your textbook 2 History of Trigonometry • The subject name is from the Greeks – Trigonon: Triangle – Metron: Measure • The Greeks developed the subject largely as a form of measurement for Astronomy ~ 3 BC • As the name indicates, it is largely the study of triangles, or perhaps the study of geometry using triangles 3 Euclid • The Greek Alexandrian, Euclid, is by far the name most commonly associated with Trigonometry • “Author” of a thirteen volume treatise The Elements – Most commonly used textbook until the 20th century (only the Bible has been published in more editions) – Every “educated” person used it, and every intellectual had a copy on his bookshelf – The book is the oldest deductive mathematics text 4 Our Class • We are going to use deductive mathematics to explore a small part of Euclid’s world • Our study of trigonometry is restricted to – Planes – Right triangles (Confining our study to right triangles is not restrictive, since any triangle can be divided into two right triangles) 5 Applications of Trigonometry • Triangulation: Measuring distances/locations – Used in satellite navigation – Geography – Astronomy • Studies of periodic waves, such as sound and light: – Acoustics – Medical imaging – Optics • Measurement – Surveying – Civil Engineering • Animation • … 6 Some Basics 7 Angle Vocabulary • Basics: – Ray – a half line – Angle – joining of two rays – Vertex – common endpoint of rays forming an angle • Special Angles: – Straight angle – 180 – Right angle – 90 – Complementary angles – sum to 90 – Supplementary angles – sum to 180 • Decimals vs. Degrees/Minutes/Seconds – 60 min in a degree – 60 sec in a minute – Converting 8 Coterminal rays • Coterminal means that the ends of the ray (vertex) are the same • Measure of an angle is from initial to terminal side (ray) – 0 is along the x axis – Positive angles are counterclockwise – Negative angles are clockwise – 360 is a full revolution • We commonly use Greek letters for angles, e.g., y x 9 Why 360??? • Around 1500 BC, Egyptians divided the day into 24 hours, though the hours varied with the seasons originally • Greek astronomers made the hours equal. • About 300 to 100 BC, the Babylonians subdivided the hour into base-60 fractions: 60 minutes in an hour and 60 seconds in a minute. • Perhaps 360 comes from an estimate of the number of days in a year? • Additionally, 360 has lots of factors 10 What are the degree angles of the four quadrants? • Q1? • Q2? • Q3? • Q4? 11 • Q1? 0 to 90 • Q2? 90 to 180 • Q3? 180 to 170 • Q4? 170 to 360 12 Special Triangles • Right Triangle: two sides form 90 angle – Longest side is the hypotenuse – Other two sides are legs • 45-45-90 triangle – Refers to the angles of the triangle – Since Pythagorean Theorem says: a2 +b2 = c2, we can easily find the length of the sides: c2 = 2x2 , c = x 2 The sides of a 45-45-90 triangle are: x, x, x 2 13 Question • How large are the angles of an equilateral triangle?? 14 Examples • The leg of a right triangle that has a 45 degree angle has length 10 inches. What is the length of the rest of the sides? 15 Solution • We know that the two legs are the same length. If the length of the legs is x, then the hypotenuse is 2 x. • The hypotenuse is 10 2 16 30-60-90 Triangle • Take an equilateral triangle and cut it in half 2x x • How long is the 3rd side? Sides are x, 3x and 2x 17 Example • The shortest side of a 30-60-90 triangle is 2 yards. How long are the rest of the sides? What is its area? 18 Solution • The shortest side of a 30-60-90 triangle is 2 yards. How long are the rest of the sides? What is its area? • Sides are x, 3x and 2x, so we have 2, 2 3 and 4 as the sides. • The area is the base x height. Here, those are the legs. The area is 3 x2 19 Summary • Vocabulary: – Ray, Angle, Vertex – Right, Straight, Complementary, Supplementary Angles • Angles: – Degrees, Minutes – Coterminal – Positive, Negative • Triangles: – Right – 45-45-90 – 30-60-90 20 Triangle Types • Right triangle – one side is 90 • Acute – all three angles < 90 • Obtuse – one angle > 90 • Equilateral – all sides equal • Isosceles – two sides equal • Scalene – no sides equal • Questions: – Can an isosceles triangle be a right triangle? – Can an equilateral triangle be a right triangle? 21 Solution • Questions: – Can an isosceles triangle be a right triangle? yes – Can an equilateral triangle be a right triangle? no 22 Properties of Triangles • The sum of all the angles of a triangle is 180 • The sum of any two sides is larger than the third • The largest angle is opposite the largest side; similarly, the smallest angle is opposite the smallest side 23 Similar Triangles • In similar triangles, corresponding angles are equal • In similar triangles, corresponding sides are proportional A C B a b c • A=a, B=b, C=c, AB/AC= ab/ac, etc. 24 For Similar Triangles • Need one of the three: Two angles the same (implies all three) Two sides proportional and the angle between them equal Three sides proportional 25 How do we use similar triangles? • Classic problem, how tall is a tree? (Boy Scout Problem) – Measure the length of the tree’s shadow – Measure the length of a shadow of something known Tree Height/Tree Shadow = Scout Height/Scout Shadow Tree Scout Scout Shadow Tree Shadow 26 Example • We have two triangles ABC and DEF. Angle A = Angle D and Angle C = Angle F. • Side b = 4 and side e = 6. If side a is 8, how long is side d? 27 Solution • We have two triangles ABC and DEF. Angle A = Angle D and Angle C = Angle F. • Side b = 4 and side e = 6. If side a is 8, how long is side d? • Since two angles are equal, all three are and the triangle are similar • Since b/e = 4/6 = 2/3, we have a/d = 2/3 = 8/12. Side d is 12 28 Summary • Types of triangles – Obtuse, Acute – Equilateral, Isosceles, Scalene • Largest side is opposite largest angle, etc. • Similar triangles – Equal angles – Proportional sides 29 Standard Position • An angle is in Standard Position if its vertex is at the origin of the axes and its initial side is along the x axis Angle 30 The Equation of the Coincident Line • Slope, m = (y2 – y1)/(x2 – x1) • But one of the points is (0,0), so the slope is y/x, where (x,y) is any point on the line Is the measure of the angle y=mx • So, given any point on the line formed by the ray, we have the slope • Note, every triangle formed by the line and the x axis is similar! 31 Example • For an angle in standard form with one ray going through the point (2,3) what is the equation of the line? • Give two other points that are on the line. 32 Solution • For an angle in standard form with one ray going through the point (2,3) what is the equation of the line? y = 3/2 x • Give two other points that are on the line. (4,6), (8, 12) 33 More Characteristics • Consider the length of the ray, r formed by the angle – r is a side of a triangle – All of the triangles formed by the ray are similar, since they have the same angles • Because similar triangles have proportional sides we know that x/r and y/r are constant for any point on the ray! • If x/r is constant, so is r/x and, therefore, r/y 34 Key Point For every point on the ray, we have: x/y, y/x, x/r constant and r/x, y/r, r/y constant! 35 Definitions In Trigonometry, we give names to these ratios: sine = y/r cosine = x/r tangent = y/x cosecant = r/y secant = r/x For x, y non-zero cotangent = x/y 36 Alternate Approach y Hypotenuse Opposite Adjacent x Sine = Opposite/Hypotenuse Cosine = Adjacent/Hypotenuse Tangent = Opposite /Adjacent SOHCAHTOA 37 Abbreviations • Cosine: cos • Sine: sin • Tangent: tan • Cosecant: csc • Secant: sec • Cotangent: cot 38 Example • If a terminal side of an angle is the line containing the point (3,4), what are the values of the trigonometric functions? – sin – cos – tan – csc – sec – cot 39 Solution • If a terminal side of an angle is the line containing the point (3,4), what are the values of the trigonometric functions? Need r = sqrt (9 + 16) = sqrt ( 25 ) = 5 – sin = 4/5 – cos = 3/5 – tan = 4/3 – csc = 5/3 – sec = 5/4 – cot = 3/4 40 Slope vs. Tangent • Slope is defined as the vertical change/ horizontal change • If one point of the line is (0, 0), the origin, the slope is the y value of the point / the x value • Therefore, the slope is just the tangent! y (x, y) Hypotenuse Opposite Adjacent x 41 More Examples • The terminal side of an angle is in the second quadrant (QII) coincident with the line y = - 12/5 x. Find sin, cos, tan • Suppose the terminal side of the angle is in QIV? What are sin, cos, tan? 42 Solutions • The terminal side of an angle is in the second quadrant (QII) coincident with the line y = - 12/5 x sin = 12/13 cos = - 5/13 tan = -12/5 • Suppose the terminal side of the angle is in QIV? the signs of the cos function is + and the sin - 43 Example • Suppose a ray starting at the origin goes through the point (2,3) in the plane, making an angle with the origin of A • What are sin A, cos A and tan A? • What is the equation of the line coincident with the ray? 44 Solution • Suppose a ray starting at the origin goes through the point (2,3) in the plane, making an angle with the origin of A • If the point is (2,3), we have the x distance 2, and y dist 3 need the hypotenuse: 4 + 9 = 13, h = 13 • Cos A = 2/ 13 , sin A = 3/ 13 , tan = 3/2 • Tan also gives the slope • y = 2/3x + b, b = 0 because goes through the origin 45 Reciprocal Identities • sin a = 1/csc a • cos a = 1/sec a • tan a = 1/cot a • csc a = 1/sin a • sec a = 1/cos a • cot a = 1/tan a 46 Signs of Trig Functions Q 2 Sin >0 Q 1 All > 0 Q 4 Cos >0 Q 3 Tan >0 ASTC – All Students Take Classes Reciprocal functions have the same sign 47 Examples: Which Quadrant? • Sin > 0, Tan < 0 ? • Cos > 0, Tan < 0 ? • Sin > 0, Tan > 0 ? • Cos > 0, Tan > 0? • Sin > 0, Cos > 0? • Csc > 0? • Cot > 0 ? 48 Solution • Sin > 0, Tan < 0 ? 2 • Cos > 0, Tan < 0 ? 4 • Sin > 0, Tan > 0 ? 1 • Cos > 0, Tan > 0? 1 • Sin > 0, Cos > 0? 1 • Csc > 0? 3, 4 • Cot > 0 ? 1, 3 49 What happens on the axes? • What are the trig functions at 0, 90, 180, and 270 deg? 50 Values of Trig Functions on Axes 0 90 180 270 Sin 0 1 0 -1 Cos 1 0 -1 0 Tan 0 undef 0 undef Sec 1 undef -1 undef Csc undef 1 undef -1 Cot undef 0 undef 0 51 Examples: Value of Functions • If cos = -5/13 and sin > 0, find the values of other functions: 52 Solution • If cos = -5/13 and sin > 0, find the values of other functions: Sin = 12/13 Tan = -12/5 Csc = -13/5 Sec = -5/12 Cot = -5/12 53 Note • We will use degrees for angles unless otherwise stated! 54 Trig Values of 45-45-90 Triangles • Sides are x, x, x 2 • Which is the largest side? • Sin 45 = 1/ 2 • Cos 45 = 1/ 2 • Tan 45 = 1 • Find csc, sec, cot for 45 deg 55 Solution • Find csc, sec, cot for 45 deg • Csc - 2 • Sec= 2 • Cot = 1 56 Trig Values of 30-60-90 Triangles • Sides are x, 2x, x 3 • Which side is smallest, largest? • cos 30 = 3/2 • sin 30 = 1/2 • tan 30 = 1/ 3 = 3/3 • cos 60 = 1/2 • sin 60 = 3/2 • tan 60 = 3 57 Example • Suppose a is an acute angle and cos a = 2/5 • Find sin a and tan a 58 Solution • Suppose a is an acute angle and cos a = 2/5 • Find sin a and tan a • Use Pythagorean Theorem 4 + x2 = 25, x = 21 • Sin a = 21/5 • Tan a = 21/2 59 What is the angle? • Cos A = 0 • Sin B = 1 • Tan C = 1 • Cot D = -1 60 Solution • Cos A = 0 A = 0, 180 • Sin B = 1 B = 90 • Tan C = 1 C = 45, 225 • Cot D = -1 D = 135, 315 61 Problems Using the values given, find the six trig functions in Quadrant I • sec a = 6/5 • tan b = 1/2 • sin d = 3/4 62 Solutions Using the values given, find the six trig functions in Quadrant I • sec a = 6/5, a2 + 25 = 36, a = 11 cos= 5/6, sin a = 11/6, tan = 11/5, cot = 5/ 11, csc= 6/ 11 • tan b = ½, 1 + 4 = 5, b = 5 Cot = 2, sin=1/ 5, cos = 2/ 5, csc = 5, sec = 5/2 • sin d = ¾, 9 + d2 = 16, d= 5 Cos = 5/4, tan = 3/ 5, sec = 4/ 5, csc = 4/3, cot = 5/3 63 Example Find the sides of a triangle with • tan b = -1/2 64 Find sides of a triangle with • tan b = -1/2 • Needs to be in quadrant 2 or 3 • Sides are 1, 2, 5 or multiples of them 65 Summary • Trigonometric Functions Defined • Understand the signs of the angles in the coordinate plane • Can evaluate trig functions in quadrants and on axes • Reciprocal Identities • 30-60-90 and 45-45-90 trig values • Using Pythagorean Theorem 66