Trigonometric Functions Chapter 2 Trigonometric Functions Page 186

Chapter 2 Overview Page 187

Chapter 2 Objectives 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. Page 187

Section 2.1 Angles in the Cartesian Plane 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. Pg 188

Angles in Standard Position 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. Page 189 We say that an angle lies in the quadrant in which its terminal side lies.

Sketching Angles in Standard Positions 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. Page 189

Coterminal angles 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. Page 193

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.

Your Turn: Measuring of 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. Page 195

Common Angles in Standard Position 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

Common Angles in Standard Position The common angles with their exact values for their Cartesian coordinates are shown on this graph. Page 198

Section 2.2 Definition 2 of Trigonometric Functions: Cartesian Plane Conceptual Objectives Define trigonometric functions in the Cartesian plane. Extend right triangle definitions of trigonometric functions for acute angles to definitions of trigonometric functions for all angles in the Cartesian plane. Understand why some trigonometric functions are undefined for quadrantal angles. Skills Objectives Calculate trigonometric function values for acute angles. Calculate trigonometric function values for nonacute angles. Calculate trigonometric function values for quadrantal angles. Page 203

The 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:

Trigonometric Functions All of the trigonometric functions are defined by the values of the three sides of a right triangle. Page 212

Calculating Trigonometric Function Values For this angle x = 2 and y = 5. The distance from the origin is . sinθ = = cosθ = = tanθ = = The remainder are calculated from these three values.

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 = . Click for answers!

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 = . sinθ = = cosθ = = tanθ = = 3

Quadrantal Values 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° 1 U 90° 180° -1 270° 360°

Section 2.3 Trigonometric Functions of Nonacute Angles Skills Objectives Determine the reference angle of a nonacute angle. Evaluate trigonometric functions exactly for common angles. Approximate trigonometric functions of nonacute angles. Conceptual Objectives Determine algebraic signs of trigonometric functions for all four quadrants. Determine values for trigonometric functions for quadrantal angles. Determine ranges for trigonometric functions. Page 216

Algebraic Signs of Trigonometric Functions 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 Pages 217-218 POSITIVE All Students Take Calculus

Using the Algebraic Sign of a Trigonometric Function 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. Page 222

Values of Quadrantal Trigonometric Functions 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. Page 223

Reference Triangle A reference triangle is formed by "dropping" a perpendicular from the terminal ray of a standard position angle to the x-axis.   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.

Section 2.4 Basic Trigonometric Identities 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. Page 230

Reciprocal 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) . Page 241

Quotient Indentities 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.

Pythagorean Identities 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