Trigonometry and Vectors Background – Trigonometry 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. http://en.wikipedia.org/wiki/History_of_trigonometry http://www.clarku.edu/~djoyce/trig/

Trigonometry and Vectors Total degrees in a triangle: Three angles of the triangle below: Three sides of the triangle below: Pythagorean Theorem: x2 + y2 = r2 a2 + b2 = c2 180 A, B, and C r, y, and x B r y HYPOTENUSE A C x

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: x2 + y2 = r2 B r y HYPOTENUSE A C x

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

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

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

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

Trigonometry and Vectors Trigonometric Functions – Sine Standard triangle labeling. Sine of <A is equal to the side opposite <A divided by the hypotenuse. sin A = opposite hypotenuse B http://en.wikipedia.org/wiki/History_of_trigonometry sin A = yr OPPOSITE r HYPOTENUSE y ADJACENT A C x

Trigonometry and Vectors Trigonometric Functions – Cosine Standard triangle labeling. Cosine of <A is equal to the side adjacent <A divided by the hypotenuse. cos A = adjacent hypotenuse B http://en.wikipedia.org/wiki/History_of_trigonometry cos A = xr OPPOSITE r HYPOTENUSE y ADJACENT A C x

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 B http://en.wikipedia.org/wiki/History_of_trigonometry tan A = yx OPPOSITE r HYPOTENUSE y ADJACENT A C x

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

Trigonometry and Vectors Trigonometric Function Problems For <A below, calculate Sine, Cosine, and Tangent: 3 4 5 B sin A = opposite hypotenuse a) 35 opposite adjacent sin A = tan A = A C 34 cos A = adjacent hypotenuse tan A = 45 cos A =

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

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

Trigonometry and Vectors Trigonometric Functions Trigonometric functions are ratios of the lengths of the segments that make up angles. sin A = opposite hypotenuse cos A = adjacent hypotenuse http://en.wikipedia.org/wiki/History_of_trigonometry tan A = opposite adjacent

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

Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS Calculate sine, cosine, and tangent for the following angles: 30o 60o 45o 12 sin 30 = 1 2 30o 60o http://en.wikipedia.org/wiki/History_of_trigonometry √3 2 cos 30 = 1 √3 tan 30 =

Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS Calculate sine, cosine, and tangent for the following angles: 30o 60o 45o √3 2 sin 60 = 1 2 30o 60o http://en.wikipedia.org/wiki/History_of_trigonometry 12 cos 60 = tan 60 = √3

Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS Calculate sine, cosine, and tangent for the following angles: 30o 60o 45o 1 45o 1 √2 cos 45 = http://en.wikipedia.org/wiki/History_of_trigonometry 1 √2 sin 45 = tan 45 = 1

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 45 degrees 90 degrees 180 degrees 270 degrees 360 degrees = -330 degrees -315 degrees -270 degrees -180 degrees -90 degrees http://en.wikipedia.org/wiki/Image:World_population.svg 180 INITIAL SIDE 270

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. http://www.chutedesign.co.uk/design/protractor/protractor.htm

Trigonometry and Vectors CLASSWORK / HOMEWORK Complete problems 1-3 on the Trigonometry Worksheet