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Trigonometry and Vectors

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 .

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Trigonometry and Vectors

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  1. Trigonometry and Vectors Background – Trigonometry • Trigonometry, triangle measure,from Greek. • Mathematics that deals with the sidesand 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.

  2. Trigonometry and Vectors Trigonometry • 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  17. Trigonometry and Vectors 60o 2 1 30o Trigonometric Functions • NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS • Calculate sine, cosine, and tangent for the following angles: • 30o • 60o • 45o 12 sin 30 = √3 2 cos 30 = 1 √3 tan 30 =

  18. Trigonometry and Vectors 60o 2 1 30o Trigonometric Functions • NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS • Calculate sine, cosine, and tangent for the following angles: • 30o • 60o • 45o √3 2 sin 60 = 12 cos 60 = tan 60 = √3

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

  20. 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 -330 degrees -315 degrees -270 degrees -180 degrees -90 degrees 30 degrees 45 degrees 90 degrees 180 degrees 270 degrees 360 degrees = = = = = 180 0 INITIAL SIDE 270

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