生物力学(三角函数).pptx

1 Basic Studying for Biomechanics 2 (Trigonometry) ( Trigonometric Ratio) (Trigonometric Function) ( Trigonometric Identities) ( Law of Sines) ( Law of Cosines) ? Contents ? vs. / ? degree VS. radian Angles add to 180 The angles of a triangle always add up to 180 44 6868 20 130 30 44 68 + 68 180 20 30 180 + 130 Right triangles We only care about right triangles A right triangle is one in which one of the angles is 90 Heres a right triangle: We call the longest side the hypotenuse We pick one of the other angles-not the right angle We name the other two sides relative to that angle Heres the right angle hypotenuse Heres the angle we are looking at adjacent opposite The Pythagorean Theorem If you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse adj2 + opp2 = hyp2 32 + 42 = 52, or 9 + 16 = 25 hyp = sqrt(adj2 + opp2) 5 = sqrt(9 + 16) 5-12-13 There are few triangles with integer sides that satisfy the Pythagorean formula 3-4-5 and its multiples (6-8-10, etc.) are the best known 5-12-13 and its multiples form another set 25 + 144 = 169 hyp adj opp Ratios Since a triangle has three sides, there are six ways to divide the lengths of the sides Each of these six ratios has a name (and an abbreviation) Three ratios are most used: sine = sin = opp / hyp cosine = cos = adj / hyp tangent = tan = opp / adj The other three ratios are redundant with these and can be ignored The ratios depend on the shape of the triangle (the angles) but not on the size hypotenuse adjacent opposite hypotenuse adjacent opposite Special Right Triangles 30 30 45 60 45 2 1 1 1 1 Using the ratios If you know the angle marked in red (call it A) and you know the length of the adjacent side, then tan A = opp / adj, so length of opposite side is given by opp = adj * tan A cos A = adj / hyp, so length of hypotenuse is given by hyp = adj / cos A hypotenuse adjacent opposite When solving oblique triangles, simply using trigonometric functions is not enough. You need The Law of Sines The Law of Cosines a2=b2+c2-2bc cosA b2=a2+c2-2ac cosB c2=a2+b2-2ab cosC It is useful to memorize these laws. They can be used to solve any triangle if enough measurements are given. a c b A B C The Six Trigonometric Ratios q q q q q q The Cosecant, Secant, and Cotangent of The Cosecant, Secant, and Cotangent of q q are the Reciprocals of are the Reciprocals of the Sine, Cosine,and Tangent of the Sine, Cosine,and Tangent of q.q. Solving a Problem with the Tangent Ratio 60 53 ft h = ? We know the angle and the We know the angle and the side adjacent to 60. We want to side adjacent to 60. We want to know the opposite side. Use theknow the opposite side. Use the tangent ratio:tangent ratio: 1 2 Why? Trigonometric Functions on a Rectangular Coordinate System x y q q Pick a point on the terminal ray and drop a perpendicular to the x-axis. (The Rectangular Coordinate Model)(The Rectangular Coordinate Model) Trigonometric Functions on a Rectangular Coordinate System x y q q Pick a point on the terminal ray and drop a perpendicular to the x-axis. r y x The adjacent side is x The opposite side is y The hypotenuse is labeled r This is called a REFERENCE TRIANGLE. Trigonometric Values for angles in Quadrants II, III and IV x y Pick a point on the terminal ray and drop a perpendicular to the x-axis. q y x r Trigonometric Values for angles in Quadrants II, III and IV x y Pick a point on the terminal ray and raise a perpendicular to the x-axis. q Trigonometric Values for angles in Quadrants II, III and IV x y Pick a point on the terminal ray and raise a perpendicular to the x-axis. q x r y Important! The is ALWAYS drawn to the x-axis Signs of Trigonometric Functions x y A A ll are positive in QI T Tan ( graph one cycle; then repeat the cycle over the interval. maxx-intminx-intmax 30-303 y = 3 cos x 20x (0, 3) ( , 0) ( , 0) ( , 3) ( , 3) 40 Amplitude The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| 1, the amplitude stretches the graph vertically. If 0 1, the amplitude shrinks the graph vertically. If a 1, the graph of the function is shrunk horizontally. y x period: 2 period: 4 42 y x y = cos (x) Graph y = f(-x) Use basic trigonometric identities to graph y = f (x) Example 1: Sketch the graph of y = sin (x). Use the identity sin (x) = sin x The graph of y = sin (x) is the graph of y = sin x reflected in the x-axis. Example 2: Sketch the graph of y = cos (x). Use the identity cos (x) = cos x The graph of y = cos (x) is identical to the graph of y = cos x. y x y = sin x y = sin (x) y = cos (x) 43 y x 0 202 0y = 2 sin 3x 0 x Example: y = 2 sin(-3x) Example: Sketch the graph of y = 2 sin (3x). Rewrite the function in the form y = a sin bx with b 0 amplitude: |a| = |2| = 2 Calculate the five key points. (0, 0) ( , 0) ( , 2) ( , -2) ( , 0) Use the identity sin ( x) = sin x: y = 2 sin (3x) = 2 sin 3x period: 22 3 = 44 The graph of y = A sin (Bx C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C/B. The number C/B is called the phase shift. amplitude = | A| period = 2 /B. x y Amplitude: | A| Period: 2/B y = A sin Bx Starting point: x = C/B The Graph of y = Asin(Bx - C) 45 Example Determine the