Trigonometry三角学.ppt

第 五 章,Trigonometry 三角,尽管三角学在ACT数学考试中所占比例不足7%，只有4或5道题，但这个知识点涉及面却很广。ACT数学考试试题可能会来自下列知识点中的一个。, Angles 角； Trigonometric Functions 三角函数； Trigonometric Identities 三角恒等式； Graphs of Trigonometric Functions 三角函数图像； Right Triangle Trigonometry 直角三角函数； Triangle Problems 三角形问题。,第一节 Angles 角,一、 Radians 弧度 Angles can be measures in degrees or in radians (abbreviated as “rad”). The angle given by a complete revolution contains 360,which is 2 rad. Therefore, 1 rad = (180/) 57.3 1= (/180) rad 0.017 rad,The following table gives the correspondence between degree and radian measures of some common angles,二、Angle in Standard Position 角的标准坐标位置,The standard position of an angle occurs when we place its vertex at the origin of a coordinate system and its initial side on the positive x-axis. The quadrant that contains the terminal side determines the quadrant that the angle lies in. In the figure above, represents an angle in Quadrant I, while is in Quadrant III. A positive angle is obtained by rotating the initial side counterclockwise until it coincides with the terminal side. Likewise, negative angles are obtained by clockwise rotation. In the figure above, is positive, while is negative.,If the terminal side of an angle in standard position is one of the axes, the angle is a quadrant angle. For example, 90(/2) and -180(-) are quadrant angles.,Every angle in standard position has a reference angle, which is the positive acute angle formed by the terminal side of the given angle and the x-axis. See examples below.,第二节 Trigonometric Functions 三角函数,For a general angle in standard position, we let P (x, y) be any point on the terminal side of and let r be the distance |OP| as shown in the figure above. Then we define the following trigonometric functions: sin =y/r csc =r/y cos =x/r sec =r/x tan =y/x cot =x/y,Notice from the diagram that is in Quadrant II, where x0 (r is always positive). Therefore, sin and csc are the only two ratios that are positive in Quadrant II. All the other ratios are negative. This is true for all Quadrant II angles.,Trig Functions of Important Angles 重要角的三角函数值,第三节 Trigonometric Identities 三角恒等式,A trigonometric identity is an equation involving trigonometric functions that holds true for all angles. Here are some of the familiar identities that you should know. 1. Quotient Identities sin =1/csc cos=1/sec cot=1/tan tan =sin /cos cot =cos /sin ,2. Pythagorean Identities Sin +cos =1 1+tan =sec 1+cot =csc 3. Periodicity Since angles and 2k(where k Z) have the terminal side, we have Sin(+2k)=sin cos(+2k)=cos,4. Symmetry Sin(-)=-sin con (-)=cos 5. Double Angle Formulas Sin2=2sincos Cos2=cos-sin=2cos-1=1-2sin 6. Sum and Difference of Two Angles Sin(+)=sincos+cossin Sin(-)=sincos-cossin Cos(+)=coscos-sinsin cos(-)=coscos+sinsin,第四节 The Graphs of Trigonometric Functions 三角函数图像,1. Periodicity 周期性 All of the trig functions are periodic, that is, f (x+p)=f (x) for all x in the domain of f, meaning the graph repeats it pattern after some interval in x. The smallest possible value of p in the expression f (x+p)=f (x) is called the fundamental period of the function, sometimes just called the period.,2. Amplitude 幅度 The sine and cosine functions have an additional property, amplitude, which is half the distance from the crest (top) to the bottom of a wave. For a sine or cosine curve that has not been vertically translated, the amplitude is simply the distance from the x-axis to the crest of the wave. The following are the graphs of the six trig functions. The domain, range, fundamental period, and amplitude (where applicable) are given for each function.,The graph of y =A sinBx and y=AcosBx Fundamental period =2/|B| Amplitude= |A| For example, the graph of the function y=4sin3x has fundamental period 2/3 and amplitude 4 The function y =-6cos1/2 x has fundamental period 4and amplitude 6.,第五节 Right Triangle Trigonometry 直角三角函数,sinx=b/c and siny=a/c cosx=a/