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UNIT 3 – ENERGY AND POWER Topics Covered HAVE– Fuels PAPER 1. Energy Sources and Power Plants PROTRACTORS/RULERS FOR 2. Trigonometry and Vectors 3. Classical Mechanics: STUDENTS WITHOUT THEM, SELL Force, Work, Energy, and Power THEM FOR $0.25 / $0.50 4. Impacts of Current Generation and Use IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Background – Trigonometry 1. 2. 3. 4. 5. 6. 7. 8. Trigonometry, triangle measure, from Greek. Mathematics that deals with the sides and angles of triangles, and their relationships. Computational Geometry (Geometry – earth measure). Deals mostly with right triangles. Historically developed for astronomy and geography. Not the work of any one person or nation – spans 1000s yrs. REQUIRED for the study of Calculus. Currently used mainly in physics, engineering, and chemistry, with applications in natural and social sciences. IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometry 1. 2. 3. 4. Total degrees in a triangle: 180 Three angles of the triangle below: A, B, and Three sides of the triangle below: a, b, and c Pythagorean Theorem: B C a2 + b2 = c2 c A b a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometry State the Pythagorean Theorem in words: “The sum of the squares of the two sides of a right triangle is equal to the square of the hypotenuse.” Pythagorean Theorem: B a2 + b2 = c2 c A b a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometry – Pyth. Thm. Problems NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 1. Solve for the unknown hypotenuse of the following triangles: a) b) ? 3 4 ? c) 1 ? 1 3 1 c2 a 2 b2 2 2 2 2 c a b c a b c a 2 b2 Align equal signs possible 2 2 2 when 2 ( 3) 1 1 1 9 16 3 1 c 2 c5 c2 Trigonometry and Vectors Common triangles in Geometry and Trigonometry 5 1 3 4 Trigonometry and Vectors Common triangles in Geometry and Trigonometry You must memorize these triangles 45o 60o 2 2 1 30o 45o 1 2 1 3 3 Trigonometry and Vectors Trigonometry – Pyth. Thm. Problems NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 2. Solve for the unknown side of the following triangles: a) 10 b) 8 ? c a b Divide all2 sides by 2 2 2 a c3-4-5 b triangle a c2 b2 2 2 2 102 82 36 13 a 6 ? c) 12 ? 12 15 a c2 b2 a c2 b2 132 122 169 144 25 a 5 Divide all2sides 15 122 by 3 3-4-5 triangle 225 144 81 a 9 Trigonometry and Vectors Trigonometric Functions – Sine Standard triangle labeling. Sine of <A is equal to the side opposite <A divided by the hypotenuse. opposite sin A = hypotenuse a sin A = c B c A ADJACENT b OPPOSITE 1. 2. a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Functions – Cosine Standard triangle labeling. Cosine of <A is equal to the side adjacent <A divided by the hypotenuse. adjacent cos A = hypotenuse cos A = b c B c A ADJACENT b OPPOSITE 1. 2. a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Functions – Tangent Standard triangle labeling. Tangent of <A is equal to the side opposite <A divided by the side adjacent <A. tan A = opposite adjacent tan A = a b B c A ADJACENT b OPPOSITE 1. 2. a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Function Problems NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 3. For <A below calculate Sine, Cosine, and Tangent: B c) b) a) B 5 3 4 C 2 B 2 1 Sketch and answer in your notebook A opp. sin A = hyp. A 1 C tan A = opp. adj. A 3 cos A = 1 C adj. hyp. Trigonometry and Vectors Trigonometric Function Problems 3. For <A below, calculate Sine, Cosine, and Tangent: a) A B 5 4 opposite sin A = hypotenuse 3 C 3 sin A = 5 adjacent cos A = hypotenuse 4 cos A = 5 opposite tan A = adjacent 3 tan A = 4 Trigonometry and Vectors Trigonometric Function Problems 3. For <A below, calculate Sine, Cosine, and Tangent: B b) A 2 1 1 C opposite sin A = hypotenuse 1 sin A = √2 adjacent cos A = hypotenuse cos A = 1 √2 opposite tan A = adjacent tan A = 1 Trigonometry and Vectors Trigonometric Function Problems 3. For <A below, calculate Sine, Cosine, and Tangent: B c) A 2 1 3 C opposite sin A = hypotenuse 1 sin A = 2 opposite tan A = adjacent adjacent cos A = hypotenuse tan A = 1 √3 cos A = √3 2 Trigonometry and Vectors Trigonometric Functions Trigonometric functions are ratios of the lengths of the segments that make up angles. opposite sin A = hypotenuse adjacent cos A = hypotenuse tan A = opposite adjacent IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Common triangles in Trigonometry You must memorize these triangles 45o 60o 2 2 1 1 30o 45o 1 3 Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4. Calculate sine, cosine, and tangent for the following angles: a. 30o 1 b. 60o sin 30 = o 2 60 c. 45o 2 cos 30 = √3 2 tan 30 = 1 √3 1 30o 3 IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4. Calculate sine, cosine, and tangent for the following angles: a. 30o √3 b. 60o sin 60 = o 2 60 c. 45o 2 1 cos 60 = 2 tan 60 = √3 1 30o 3 IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4. Calculate sine, cosine, and tangent for the following angles: a. 30o b. 60o 45o 1 cos 45 = o √2 c. 45 2 1 sin 45 = √2 tan 45 = 1 1 45o 1 IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Measuring Angles Unless otherwise specified: • Positive angles measured counter-clockwise from the horizontal. • Negative angles measured clockwise from the horizontal. • We call the horizontal line 0o, or the initial side 90 30 degrees = -330 degrees 45 degrees = -315 degrees 90 degrees = -270 degrees 180 INITIAL SIDE 0 180 degrees = -180 degrees 270 degrees = -90 degrees IOT 270 360 degrees 3-8 POLY ENGINEERING Trigonometry and Vectors • • • • Begin all lines as light construction lines! Draw the initial side – horizontal line. From each vertex, precisely measure the angle with a protractor. Measure 1” along the hypotenuse. Using protractor, draw vertical line from the 1” point. Darken the triangle. Trigonometry and Vectors CLASSWORK / HOMEWORK Complete problems 1-3 on the Trigonometry Worksheet