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Trigonometry and Vectors 1.Trigonometry, triangle measure, from Greek. 2.Mathematics that deals with the sides and angles of triangles, and their relationships. 3.Computational Geometry (Geometry – earth measure). 4.Deals mostly with right triangles. 5.Historically developed for astronomy and geography. 6.Not the work of any one person or nation – spans 1000s yrs. 7.REQUIRED for the study of Calculus. 8.Currently used mainly in physics, engineering, and chemistry, with applications in natural and social sciences. Background – Trigonometry

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Trigonometry and Vectors 1.Total degrees in a triangle: 2.Three angles of the triangle below: 3.Three sides of the triangle below: 4.Pythagorean Theorem: x 2 + y 2 = r 2 a 2 + b 2 = c 2 Trigonometry 180 A B C r, y, and x y x r HYPOTENUSE A, B, and C

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Trigonometry and Vectors 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: x 2 + y 2 = r 2 Trigonometry A B C y x r HYPOTENUSE

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Trigonometry and Vectors NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 1.Solve for the unknown hypotenuse of the following triangles: Trigonometry – Pyth. Thm. Problems 4 3 ? a) 1 1 ? b) 1 ? c) Align equal signs when possible

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Trigonometry and Vectors Common triangles in Geometry and Trigonometry 3 4 5 1

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Trigonometry and Vectors Common triangles in Geometry and Trigonometry 1 1 1 45 o 2 30 o 60 o You must memorize these triangles 2 3

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Trigonometry and Vectors NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 2.Solve for the unknown side of the following triangles: Trigonometry – Pyth. Thm. Problems 8 ? 10 ? 15 ? 12 13 12 a) b) c) Divide all sides by 2 3-4-5 triangle Divide all sides by 3 3-4-5 triangle

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Trigonometry and Vectors 1.Standard triangle labeling. 2.Sine of <A is equal to the side opposite <A divided by the hypotenuse. Trigonometric Functions – Sine A B C y x r HYPOTENUSE OPPOSITE ADJACENT sin A = yryr opposite hypotenuse

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Trigonometry and Vectors 1.Standard triangle labeling. 2.Cosine of <A is equal to the side adjacent <A divided by the hypotenuse. Trigonometric Functions – Cosine A B C y x r HYPOTENUSE OPPOSITE ADJACENT cos A = xrxr adjacent hypotenuse

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Trigonometry and Vectors 1.Standard triangle labeling. 2.Tangent of <A is equal to the side opposite <A divided by the side adjacent <A. Trigonometric Functions – Tangent A B C y x r HYPOTENUSE OPPOSITE ADJACENT tan A = yxyx opposite adjacent

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Trigonometry and Vectors 3 4 5 1 2 1 1 NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 3.For <A below calculate Sine, Cosine, and Tangent: Trigonometric Function Problems A B C A B C A B C a) b) c) sin A = opp. hyp. cos A = adj. hyp. tan A = opp. adj. Sketch and answer in your notebook

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Trigonometry and Vectors 3 4 5 3.For <A below, calculate Sine, Cosine, and Tangent: Trigonometric Function Problems A B C a) sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent sin A = 3535 cos A = 4545 tan A = 3434

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Trigonometry and Vectors 3.For <A below, calculate Sine, Cosine, and Tangent: Trigonometric Function Problems sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent sin A = 1 √2 cos A = tan A = 1 1 1 A B C b) 1 √2

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Trigonometry and Vectors 3.For <A below, calculate Sine, Cosine, and Tangent: Trigonometric Function Problems sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent sin A = 1212 cos A = tan A = √3 2 1 2 A B C c) 1 √3

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Trigonometry and Vectors Trigonometric functions are ratios of the lengths of the segments that make up angles. Trigonometric Functions tan A = opposite adjacent sin A = opposite hypotenuse cos A = adjacent hypotenuse

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Trigonometry and Vectors Common triangles in Trigonometry 1 1 45 o 1 2 30 o 60 o You must memorize these triangles

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Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4.Calculate sine, cosine, and tangent for the following angles: a.30 o b.60 o c.45 o 1 2 30 o 60 o sin 30 = 1212 cos 30 = √3 2 tan 30 = 1 √3

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Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4.Calculate sine, cosine, and tangent for the following angles: a.30 o b.60 o c.45 o 1 2 30 o 60 o cos 60 = 1212 sin 60 = √3 2 tan 60 = √3

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Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4.Calculate sine, cosine, and tangent for the following angles: a.30 o b.60 o c.45 o tan 45 = 1 sin 45 = 1 √2 cos 45 = 1 √2 1 1 45 o

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Unless otherwise specified: Positive angles measured counter-clockwise from the horizontal. Negative angles measured clockwise from the horizontal. We call the horizontal line 0 o, or the initial side 0 90 180 270 Trigonometry and Vectors Measuring Angles 30 degrees 45 degrees 90 degrees 180 degrees 270 degrees 360 degrees INITIAL SIDE -330 degrees -315 degrees -270 degrees -180 degrees -90 degrees ==========

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