Trigonometry and Vectors Background Trigonometry 1 2 3

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 1000 s yrs. REQUIRED for the study of Calculus. Currently used mainly in physics, engineering, and chemistry, with applications in natural and social sciences.

Trigonometry and Vectors Trigonometry 1. 2. 3. 4. 5. 6. Total degrees in a triangle: 180 Three angles of the triangle below: A, B, and Three sides of the triangle below: r, y, and x Pythagorean Theorem: B x 2 + y 2 = r 2 a 2 + b 2 = c 2 A E OT P HY E S U N r x y C C

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 x 2 + y 2 = r 2 HY A E S U N r E OT P x y C

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

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 45 o 2 1 45 o 2 60 o 1 3

Trigonometry and Vectors Trigonometry – Pyth. Thm. Problems NO CALCULATORS – SKETCH 2. – SIMPLIFY ANSWERS Solve for the unknown side of the following triangles: a) 10 8 ? Divide all sides by 2 3 -4 -5 triangle b) 13 12 ? c) 12 ? 15 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. opposite sin A = hypotenuse y sin A = r B E T O P Y H A E S U N r ADJACENT x OPPOSITE 1. 2. y C

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 = x r B E T O P Y H A E S U N r ADJACENT x OPPOSITE 1. 2. y C

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 = y x B E T O P Y H A E S U N r ADJACENT x OPPOSITE 1. 2. y C

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

Trigonometry and Vectors Trigonometric Function Problems 3. For <A below, calculate Sine, Cosine, and Tangent: a) A B 5 4 3 C opposite sin A = hypotenuse 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 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: c) A B 2 1 C opposite sin A = hypotenuse 1 sin A = 2 opposite tan A = 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

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

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

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

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 45 o 1 cos 45 = o √ 2 c. 45 1 sin 45 = √ 2 tan 45 = 1 1 45 o 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 0 o, or the initial side 90 30 degrees = -330 degrees 45 degrees = -315 degrees 180 INITIAL SIDE 0 90 degrees = -270 degrees 180 degrees = -180 degrees 270 degrees = -90 degrees 270 360 degrees

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