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Welcome to Week 1 College Trigonometry Trigonometry? What is trigonometry? Trigonometry? Tri means three Gon means sides Ometry means measurement Trigonometry? Originated with the Egyptians Trigonometry? Arabic Trigonometry was developed in order to observe holy days on the correct days in all parts of the Islamic world Trigonometry? There was also a need for nonnavigators to be able to travel to Mecca each year and return successfully Trigonometry? In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī wrote the first modern trig book (blame him…) Trigonometry? The trig functions and natural exponentials and logarithms we study in this class are needed in most formulas used to describe how our complicated universe works Trigonometry? astronomy banking electronics biology atoms forensics construction ... Trigonometry! At last! Something USEFUL!!! Questions? Angles To study triangles, let’s start with one corner: an angle Angles A ray is half of a line it has an origin the other end stretches on forever origin Sun rays Death rays Angles Real-life death rays: Gamma rays Cosmic rays Angles If two rays start at the same origin, they form an “angle” Angles Their point of common origin is called the “vertex” vertex Angles Angles are usually represented by lowercase Greek letters: α, β, θ θ Angles Angles have an initial (beginning) side and a terminal (ending) side terminal θ initial Angles The “standard position” for an angle: vertex at origin and initial side along x-axis θ (0,0) x Angles Positive angles counterclockwise rotation from initial side θ Angles Negative angles - clockwise rotation from initial side θ ANGLES IN-CLASS PROBLEMS Which of the following graphs represent negative angles? Angles Remember graph paper? Angles Graph paper is made of angles: Angles Graph paper is split into 4 quadrants: Angles The quadrants move counterclockwise around the grid II just like positive angles III I IV Angles An angle with a terminal side in a quadrant "lies" in that I II quadrant III IV ANGLES IN-CLASS PROBLEMS Which quadrant does each angle lie? Angles An angle with a terminal side along any axis is called “quadrantal” Angles Angles are measure by determining the amount of rotation from the initial side to the terminal side θ Angles A degree, symbolized by ° measures angles θ 23° ANGLES IN-CLASS PROBLEMS Which of the graphs represents the angle θ = –45º in standard position? Angles Types of angles: ANGLES IN-CLASS PROBLEMS Classify the angle: ANGLES IN-CLASS PROBLEMS Classify the angle: 168º 42º 90º 180º 263º Angles A “reference angle” is a positive acute angle θ° formed by the terminal side of a nonacute angle θ and the x-axis ANGLES IN-CLASS PROBLEMS Which is the reference angle? ANGLES IN-CLASS PROBLEMS Co-terminal angle θ ± k•360° Which angle is co-terminal to 275º? a) 85º c) –55º b) –85º d) 55º ANGLES IN-CLASS PROBLEMS Complementary - two positive angles whose sum = 90° Find the complementary angle to 32º a) 148º c) 328º b) –32º d) 58º ANGLES IN-CLASS PROBLEMS Supplementary - two positive angles whose sum = 180° Find the supplementary angle to 32º: a) 148º c) 328º b) –32º d) 58º ANGLES IN-CLASS PROBLEMS Explementary - two positive angles whose sum = 360° Find the explementary angle to 32º: a) 148º c) 328º b) –32º d) 58º Questions? RIGHT ANGLE TRIANGLES IN-CLASS PROBLEMS A polygon with three sides and three angles is called _______ Right Angle Triangles A triangle! RIGHT ANGLE TRIANGLES IN-CLASS PROBLEMS A special triangle has one corner that is a right angle – what is it called? RIGHT ANGLE TRIANGLES IN-CLASS PROBLEMS The side of the triangle opposite of the right angle is called: __________ Right Angle Triangles RIGHT ANGLE TRIANGLES IN-CLASS PROBLEMS Given the lengths of two sides of a right triangle, you can calculate the third using ____ a2 + b2 = c2 Right Angle Triangles Pythagorean Theorem video RIGHT ANGLE TRIANGLES IN-CLASS PROBLEMS Find the missing side: a = 4 b = 3 c = _____ a = _____ b = 3 c = 5 a = 4 b = _____ c = 5 a = 6 b = 2 c = _____ a = 1 b = 2 c = _____ a = 2 b = 1 c = _____ Questions? Trigonometry Based on a right triangle: Trigonometry Trigonometry came to us from the ancient Greeks who drew triangles in circles Trigonometry There are six trig functions: sine = length of side opposite θ length of hypotenuse Trigonometry cosine = length of side adjacent θ length of hypotenuse Trigonometry tangent = length of side opposite θ length of side adjacent θ Trigonometry cotangent = length of side adjacent θ length of side opposite θ secant = length of hypotenuse length of side adjacent θ cosecant = length of hypotenuse length of side opposite θ Trigonometry US vs Europe bakerfamilytree.blogspot.com Trigonometry Trigonometry Trigonometry Based on the formula a2+b2=c2, where c is the hypotenuse and a and b are the other sides, there are three identities: sin2 θ + cos2 θ = 1 1 + tan2 θ = sec2 θ 1 + cot2 θ = csc2 θ Trigonometry Hipparchus calculated the first trig table Trigonometry Cot/Sec/Csc Stickers TRIGONOMETRY IN-CLASS PROBLEMS Find these trig functions using your calculator: sin 10º sin 90º cos 45º tan 0º sin 0º cos 20º cos 90º tan 45º sin 45º cos 0º tan 30º tan 90º TRIGONOMETRY IN-CLASS PROBLEMS Evaluate tan 30º a) 𝟑 c) –1 b) d) 1 𝟐 TRIGONOMETRY IN-CLASS PROBLEMS Evaluate sec 45º a) 𝟑 c) –1 b) d) 1 𝟐 TRIGONOMETRY IN-CLASS PROBLEMS Given sin θ = 3/5 and cos θ = 4/5, find tan θ a) 4/5 c) 5/3 b) 3/4 d) 5/4 TRIGONOMETRY IN-CLASS PROBLEMS Given sin θ = 2/3 and cos θ = 5/3, find cot θ a) 1 b) 5/2 c) 2/ 5 d) 3/2 TRIGONOMETRY IN-CLASS PROBLEMS Given sin θ = 3/5 and cos θ = 4/5, find csc θ a) 4/5 c) 5/3 b) 3/4 d) 5/4 TRIGONOMETRY IN-CLASS PROBLEMS Given sin θ = 2/3 and cos θ = 5/3, find sec θ a) 1 c) 5/3 b) 3/5 d) 5/2 TRIGONOMETRY IN-CLASS PROBLEMS Which of the following expressions represents the same value as csc 35º? a) sec 55º c) tan 55º b) cos 55º d) cot 55º TRIGONOMETRY IN-CLASS PROBLEMS Simplify sin2 10º + cos2 10º a) 2 c) 3 b) 1 d) 10 Trigonometry TRIGONOMETRY HOMEWORK PROBLEM The wife of the victim said that she had just asked him for a divorce when he suddenly pulled a .357 out of his jacket pocket and shot himself in the head. TRIGONOMETRY HOMEWORK PROBLEM According to her statement, he was standing beside the kitchen sink at the time of the shot. The victim has a single, nearcontact entrance wound above his right ear 67” above the heel. The shot did not exit. TRIGONOMETRY HOMEWORK PROBLEM Determine the height of impact to determine if the wife is telling the truth or not. TRIGONOMETRY HOMEWORK PROBLEM Blood spatter pattern: A B C and D are the angles of impact, the other numbers are the distance to the point of convergence TRIGONOMETRY HOMEWORK PROBLEM Calculate the height of impact using the following equation: tangent of the distance from angle of impact * base of spatter to θ pt. of convergence height of = impact TRIGONOMETRY HOMEWORK PROBLEM Angle STAIN of Impact θ tan(θ) Distance (D) to Point of Convergence A 62º 20.5" B 42º 34.5" C 29º 51.375" D 22º 76.25" (D) * tan(θ) TRIGONOMETRY HOMEWORK PROBLEM 1) Sum of (D)*tan(θ) = _____ 2) Number of Stains = _____ 3) Height of impact = 1)/2) = _____ TRIGONOMETRY HOMEWORK PROBLEM If the victim's ear was 67” above the floor, was the wife telling the truth? Questions? Liberation! Be sure to turn in your exercises to me before you leave Don’t forget your lab homework due next week! Have a great rest of the week!