5 Trigonometric Identities Copyright © 2009 Pearson Addison-Wesley 5.1-1 Prayer Let us remember that we are in the Most Holy Presence of God. In the name of the Father, and of the Son and of the Holy Spirit, amen. St John Baptiste de La Salle, pray for us. Leave Jesus in our hearts, forever. SCHEDULE DATE October 12 October 13 October 14 October 15 October 19 ACTIVITY TRIGONOMETRIC IDENTITIES October 20 October 21 October 22 ONLINE QUIZ (TRIGONOMETRIC IDENTITIES )730-930A RECOLLECTION BREATHER October 26 October 27 October 28-30 November 2 November 3 November 4 TRIGONOMETRIC EQUATIONS & OBLIQUE TRIANGLES MODULE 6 PROBLEM SET LA SALLIAN DAYS HOLIDAY MODULE 6 PROBLEM SET (DISCUSSION) ONLINE QUIZ (MODULE 6 )1030-1230P November 9-12 MIDTERM EXAMS MODULE 5 PROBLEM SET MODULE 5 PROBLEM SET (DISCUSSION) 5.1-3 5 Targeted Outcomes 1.To demonstrate that the trigonometric functions of angles are related to each other through trigonometric identities. 2. Demonstrate the evaluation and graph sketching of inverse trigonometric functions. Learning Objectives 1. Familiarize with the different trigonometric identities. 2. Perform the evaluation of inverse trigonometric identities. 3. Sketching the graph of inverse trigonometric functions. Copyright © 2009 Pearson Addison-Wesley 5.1-4 5 Trigonometric Identities 5.1 Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Sum and Difference Identities for Cosine 5.4 Sum and Difference Identities for Sine and Tangent 5.5 Double-Angle Identities 5.6 Half-Angle Identities 5.7 Graphs of Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 5.1-5 “All Students Take Calculus” Note In trigonometric identities, θ can be an angle in degrees, an angle in radians, a real number, or a variable. Copyright © 2009 Pearson Addison-Wesley 1.1-6 5.1-6 FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Example 1 and θ is in quadrant II, find each function If value. c 5 q -3 Copyright © 2009 Pearson Addison-Wesley 1.1-7 5.1-7 Fundamental Identities Reciprocal Identities Quotient Identities Copyright © 2009 Pearson Addison-Wesley 1.1-8 5.1-8 Fundamental Identities Pythagorean Identities Negative-Angle Identities Copyright © 2009 Pearson Addison-Wesley 1.1-9 5.1-9 Proving Trigonometric Identities Suggestions... 1. Learn well the formulas given above (or at least, know how to find them quickly). The better you know the basic identities, the easier it will be to recognise what is going on in the problems. 2. Work on the most complex side and simplify it so that it has the same form as the simplest side. 3. Don't assume the identity to prove the identity. This means don't work on both sides of the equality sign and try to meet in the middle. Start on one side and make it look like the other side. 4. Many of these come out quite easily if you express everything on the most complex side in terms of sine and cosine only. 5. In most examples where you see power 2 (that is, 2), it will involve using the identity sin2 θ + cos2 θ = 1 (or one of the other 2 formulas that we derived above). Copyright © 2009 Pearson Addison-Wesley 1.1-10 5.1-10 Example 2 PROVING IDENTITIES Prove the following trigonometric equations are identities: Note: LHS means “Left Hand Side” and RHS means “Right Hand Side” of the equation. Copyright © 2009 Pearson Addison-Wesley 1.1-11 5.1-11 Example 3 PROVING IDENTITIES Prove the following trigonometric equations are identities: Note: LHS means “Left Hand Side” and RHS means “Right Hand Side” of the equation. Copyright © 2009 Pearson Addison-Wesley 1.1-12 5.1-12 Caution When working with trigonometric expressions and identities, be sure to write the argument of the function. For example, we would not write An argument such as θ is necessary. Copyright © 2009 Pearson Addison-Wesley 1.1-13 5.1-13 Double-Angle Identities We can use the cosine sum identity to derive double-angle identities for cosine. Cosine sum identity Copyright © 2009 Pearson Addison-Wesley 5.5-14 Double-Angle Identities There are two alternate forms of this identity. Copyright © 2009 Pearson Addison-Wesley 5.5-15 Double-Angle Identities We can use the sine sum identity to derive a double-angle identity for sine. Sine sum identity Copyright © 2009 Pearson Addison-Wesley 5.5-16 Double-Angle Identities We can use the tangent sum identity to derive a double-angle identity for tangent. Tangent sum identity Copyright © 2009 Pearson Addison-Wesley 5.5-17 Double-Angle Identities Copyright © 2009 Pearson Addison-Wesley 1.1-18 5.5-18 FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ Example 4 and sin θ < 0, find sin 2θ, cos 2θ, and Given tan 2θ. The identity for sin 2θ requires sin θ. 3 q 5 -4 - Copyright © 2009 Pearson Addison-Wesley Any of the three forms may be used. 1.1-19 5.5-19 Example 4 FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ (cont.) Alternatively, find tan 2θ by finding the quotient of sin 2θ and cos 2θ. Copyright © 2009 Pearson Addison-Wesley 1.1-20 5.5-20 Example 5 VERIFYING A DOUBLE-ANGLE IDENTITY Verify that is an identity. Quotient identity Double-angle identity Copyright © 2009 Pearson Addison-Wesley 1.1-21 5.5-21 Product-to-Sum Identities The identities for cos(A + B) and cos(A – B) can be added to derive a product-to-sum identity for cosines. Copyright © 2009 Pearson Addison-Wesley 5.5-22 Product-to-Sum Identities Similarly, subtracting cos(A + B) from cos(A – B) gives a product-to-sum identity for sines. Copyright © 2009 Pearson Addison-Wesley 5.5-23 Product-to-Sum Identities Using the identities for sin(A + B) and sine(A – B) gives the following product-to-sum identities. Copyright © 2009 Pearson Addison-Wesley 5.5-24 Product-to-Sum Identities Copyright © 2009 Pearson Addison-Wesley 1.1-25 5.5-25 Example 6 USING A PRODUCT-TO-SUM IDENTITY Write 4 cos 75° sin 25° as the sum or difference of two functions. 4{ ½ [sin(750 + 250)-sin(750 – 250)]} Click to add text Copyright © 2009 Pearson Addison-Wesley 1.1-26 5.5-26 Sum-to-Product Identities Copyright © 2009 Pearson Addison-Wesley 1.1-27 5.5-27 Half-Angle Identities We can use the cosine sum identities to derive halfangle identities. Choose the appropriate sign depending on the quadrant of Copyright © 2009 Pearson Addison-Wesley 5.6-28 Half-Angle Identities Choose the appropriate sign depending on the quadrant of Copyright © 2009 Pearson Addison-Wesley 5.6-29 Half-Angle Identities There are three alternative forms for Copyright © 2009 Pearson Addison-Wesley 5.6-30 Half-Angle Identities Copyright © 2009 Pearson Addison-Wesley 5.6-31 Double-Angle Identities Copyright © 2009 Pearson Addison-Wesley 1.1-32 5.6-32 Example 7 USING A HALF-ANGLE IDENTITY TO FIND AN EXACT VALUE Find the exact value of cos 15° using the half-angle identity for cosine. Choose the positive square root. Copyright © 2009 Pearson Addison-Wesley 1.1-33 5.6-33 Example 8 VERIFYING AN IDENTITY Verify that Copyright © 2009 Pearson Addison-Wesley is an identity. 1.1-34 5.6-34 5.7 Graphs of Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 1.1-35 5.1-35 Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 3 x 0 2 2 sin x 0 2 1 0 -1 0 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y = sin x y 3 2 - - 1 2 2 3 2 2 5 2 x -1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 36 Arcsine Function Arcsine function is an inverse of the sine function denoted by sin-1x. It is represented in the graph as shown below: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 37 Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 3 x 0 2 2 cos x 1 2 0 -1 0 1 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y = cos x y 3 2 - - 1 2 2 3 2 2 5 2 x -1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 38 Arccosine Function Arccosine function is the inverse of the cosine function denoted by cos-1x. It is represented in the graph as shown below: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 39 Graph of the Tangent Function sin x To graph y = tan x, use the identity tan x = . cos x At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes. y Properties of y = tan x 1. domain : all real x x k + (k ) 2 2. range: (–, +) 3. period: 4. vertical asymptotes: x = k + (k ) 2 2 - 3 2 3 2 x - 2 period: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 40 Arctangent Function Arctangent function is the inverse of the tangent function denoted by tan-1x. It is represented in the graph as shown below: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 41 Graph of the Cotangent Function cos x To graph y = cot x, use the identity cot x = . sin x At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes. y Properties of y = cot x y = cot x 1. domain : all real x x k (k ) 2. range: (–, +) 3. period: 4. vertical asymptotes: x = k (k ) vertical asymptotes Copyright © by Houghton Mifflin Company, Inc. All rights reserved. - 3 2 - 3 2 2 2 - x = - x=0 x= x 2 x = 2 42 Arccotangent (Arccot) Function Arccotangent function is the inverse of the cotangent function denoted by cot-1x. It is represented in the graph as shown below: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 43 Graph of the Secant Function The graph y = sec x, use the identity sec x = 1 . cos x At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes. y y = sec x Properties of y = sec x 1. domain : all real x x k + ( k ) 2 2. range: (–,–1] [1, +) 3. period: 4. vertical asymptotes: x = k + (k ) 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 y = cos x x - 2 2 3 2 2 5 3 2 -4 44 Arcsecant Function Arcsecant function is the inverse of the secant function denoted by sec-1x. It is represented in the graph as shown below: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 45 Graph of the Cosecant Function 1 To graph y = csc x, use the identity csc x = . sin x At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes. y Properties of y = csc x 4 y = csc x 1. domain : all real x x k (k ) 2. range: (–,–1] [1, +) 3. period: 4. vertical asymptotes: x = k (k ) where sine is zero. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. x - 2 2 3 2 2 5 2 y = sin x -4 46 Arccosecant Function What is arccosecant (arccsc x) function? Arccosecant function is the inverse of the cosecant function denoted by cosec -1x. It is represented in the graph as shown below: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 47 Copyright © 2009 Pearson Addison-Wesley 1.1-48 5.5-48