Trigonometric Functions Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Chapter 2 Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Chapter 2 Overview Draw angles in the Cartesian plane. Define trigonometric functions as ratios of x and y coordinates and distances in the Cartesian plane. Evaluate trigonometric functions for nonacute angles. Determine ranges for trigonometric functions and signs for trigonometric functions in each quadrant. Derive and use basic trigonometric identities. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Chapter 2 Objectives Skills Objectives Plot angles in standard position. Identify coterminal angles. Graph common angles. Conceptual Objectives Relate the x and y coordinates to the legs of a right triangle. Derive the distance formula from the Pythagorean Theorem. Connect angles with quadrants. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Section 2.1 Angles in the Cartesian Plane An angle is said to be in standard position if its initial side is along the positive x-axis and its vertex is at the origin. We say that an angle lies in the quadrant in which its terminal side lies. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Angles in Standard Position Sketching a 210º angle in the standard position yields this graph. •The initial side lies on the x-axis. •The positive angle indicates counterclockwise rotation. •180º represents a straight angle and the additional 30º yields a 210 º angle. •The terminal side lies in quadrant III. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Sketching Angles in Standard Positions Two angles in standard position with the same terminal side are called coterminal angles. For example, -40º and 320º are coterminal angles. Moving 40º in clockwise direction brings the terminal side to the same position as moving 320º in the counter-clockwise direction. Such angles may also be reached by going the same direction, such as 90º and 450º. 450º is reached by moving counterclockwise through the full 360º circle, then continuing another 90 º. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Coterminal angles If you graph angles x = 30o and y = - 330o in standard position, these angles will have the same terminal side. See figure below Coterminal angles Ac to angle A may be obtained by adding or subtracting k*360 degrees or k* (2π). Hence Ac = A + k*360o if A is given in degrees. Or Ac = A + k*(2π) if A is given in radians; where k is any negative or positive integer. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Coterminal Angles Determine the smallest possible measure of these angles: 580º Solution: Subtract 360º to find the correct angle of 220º. -400º Solution: Add 360º to get -40º. Add 360º again to get the correct angle of 320º. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Your Turn: Measuring of Coterminal Angles Standard Position: An angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is called the initial side and the other ray is called the terminal side. If the terminal side of an angle lies "on" the axes (such as 0º, 90º, 180º, 270º, 360º ), it is called a quadrantal angle. The angle shown at the right is referred to as a Quadrant II angle since its terminal side lies in Quadrant II. If Θ is an angle in standard position, and P is any point (other than the origin) on the terminal side of Θ, then we associate 3 numbers with the point P. x: x-coordinate of the point P y: y-coordinate of the point P r : distance of the point from the origin Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Common Angles in Standard Position The common angles with their exact values for their Cartesian coordinates are shown on this graph. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Common Angles in Standard Position Skills Objectives Conceptual Objectives Calculate trigonometric function Define trigonometric functions in the Cartesian plane. values for acute angles. Calculate trigonometric function Extend right triangle definitions of trigonometric functions for values for nonacute angles. acute angles to definitions of Calculate trigonometric function trigonometric functions for all values for quadrantal angles. angles in the Cartesian plane. Understand why some trigonometric functions are undefined for quadrantal angles. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Section 2.2 Definition 2 of Trigonometric Functions: Cartesian Plane Line up a right triangle with a perpendicular segment connecting the point (x, y) to the x-axis. The distance from the origin, (0, 0), to the point (x, y) is now: r ( x 0) 2 ( y 0) 2 x 2 y 2 Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. The Cartesian Plane All of the trigonometric functions are defined by the values of the three sides of a right triangle. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Trigonometric Functions For this angle x = 2 and y = 5. The distance from the origin is 29 . y sinθ = r = 5 29 2 x cosθ = = 29 r tanθ = y = 5 x 2 The remainder are calculated from these three values. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Calculating Trigonometric Function Values Calculate the values of x, y, and r in the same way. r must be positive. For this graph x = -1, y = -3, and r= 10 . Click for answers! Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Your Turn : Calculating Trigonometric Functions for Nonacute Angles Calculate the values of x, y, and r in the same way. r must be positive. For this graph x = -1, y = -3, and r= 10 . sinθ = 3 y = r 10 cosθ = x r tanθ = y =3 x = 1 10 Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Your Turn : Calculating Trigonometric Functions for Nonacute Angles The table below summarizes the trigonometric function values for common quadrantal angles: 0°, 90 °, 180 °, 270 °, and 360 °. Θ SINΘ COSΘ TANΘ COTΘ SECΘ CSCΘ 0° 0 1 0 U 1 U 90° 1 0 U 0 U 1 180° 0 -1 0 U -1 U 270° -1 0 U 0 U -1 360° 0 1 0 U 1 U Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Quadrantal Values Skills Objectives Conceptual Objectives Determine algebraic signs of Determine the reference trigonometric functions for all angle of a nonacute four quadrants. angle. Determine values for Evaluate trigonometric trigonometric functions for functions exactly for quadrantal angles. common angles. Approximate trigonometric Determine ranges for trigonometric functions. functions of nonacute angles. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Section 2.3 Trigonometric Functions of Nonacute Angles Sin Θ = y/r Cos Θ = x/r Tan Θ = y/x Csc Θ = r/y , y≠ 0 Sec Θ = r/x , x ≠ 0 Cot Θ = x/y , y ≠ 0 POSITIVE All Students Take Calculus y ° r Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Algebraic Signs of Trigonometric Functions If cosθ = -3/5 and the terminal side of the angle lies in quadrant III, find sinθ. cosθ = -3/5 means that the x value is negative, so x = -3 and r = 5. Now we know that (-3)2 + y2 = 52. y2 = 25 – 9 = 16, so y = ±4. Since the angle is in quadrant III, y = -4. sinθ = y/r = -4/5. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Using the Algebraic Sign of a Trigonometric Function The values of the trigonometric functions for angles along the axes are undefined for some angles. For example, along the positive y-axis, the value of x is zero, making the value of the tangent undefined. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Values of Quadrantal Trigonometric Functions A reference triangle is formed by "dropping" a perpendicular from the terminal ray of a standard position angle to the xaxis. Remember, it must be drawn to the x-axis. Reference triangles are used to find trigonometric values for their standard position angles. They are of particular importance for standard position angles whose terminal sides reside in quadrants II, III and IV. A reference triangle contains a reference angle. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Reference Triangle Skills Objectives Learn the reciprocal identities. Learn the quotient identities. Learn the Pythagorean identities. Use the basic identities to simplify expressions. Conceptual Objectives Understand that trigonometric reciprocal identities are not always defined. Understand that quotient identities are not always defined. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Section 2.4 Basic Trigonometric Identities Since sinθ = y/r and cscθ = r/y, these two trigonometric functions are reciprocals of one another. Therefore, if y ≠ 0, then cscθ is defined. Similarly, cosθ = x/r and secθ = r/x(defined if x ≠ 0) are reciprocal functions as are tanθ = y/x (defined if x ≠ 0) and cotθ = x/y (defined if y ≠ 0) . Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Reciprocal Identities Since tanθ = sinθ /cosθ and cotθ = cosθ /sinθ, these two trigonometric functions are called quotient identities. Therefore, if cosθ ≠ 0, then tanθ is defined and if sinθ ≠ 0, then cotθ is defined. Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Quotient Indentities When studying the unit circle, it was observed that a point on the unit circle (the vertex of the right triangle) can be represented by the coordinates (cos Θ, sin Θ ). Since the legs of the right triangle in the unit circle have the values of cos Θ and sin Θ, the Pythagorean Theorem can be used to obtain …. . Pythagorean Identities Variations Sin Θ 2 + Cos Θ 2 = 1 Sin Θ 2 = 1 - Cos Θ 2 Cos Θ 2 = 1 - Sin Θ 2 Tan Θ 2 + 1 = Sec Θ 2 Tan Θ 2 = Sec Θ 2 - 1 1 + Cot Θ 2 = Csc Θ 2 Cot Θ 2 = Csc Θ 2 - 1 Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved. Pythagorean Identities