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(Chapter 1: Review) 1.31 TOPIC 3: TRIGONOMETRY II PART A: FUNDAMENTAL TRIGONOMETRIC IDENTITIES Memorize these in both “directions” (i.e., left-to-right and right-to-left). Reciprocal Identities 1 sin x 1 sec x = cos x 1 cot x = tan x 1 csc x 1 cos x = sec x 1 tan x = cot x csc x = sin x = WARNING 1: Remember that the reciprocal of sin x is csc x , not sec x . TIP 1: We informally treat “0” and “undefined” as reciprocals when we are dealing with basic trigonometric functions. Your algebra teacher will not want to hear this, though! Quotient Identities tan x = sin x cos x and cot x = cos x sin x Pythagorean Identities sin 2 x + cos 2 x = 1 1 + cot x = csc 2 x 2 tan 2 x + 1 = sec 2 x TIP 2: The second and third Pythagorean Identities can be obtained from the first by dividing both of its sides by sin 2 x and cos 2 x , respectively. TIP 3: The squares of csc x and sec x , which have “Up-U, Down-U” graphs, are all alone on the right sides of the last two identities. They can never be 0 in value. (Why is that? Look at the left sides.) (Chapter 1: Review) 1.32 Cofunction Identities If x is measured in radians, then: sin x = cos x 2 cos x = sin x 2 We have analogous relationships for tangent and cotangent, and for secant and cosecant; remember that they are sometimes undefined. Think: Cofunctions of complementary angles are equal. Even / Odd (or Negative Angle) Identities Among the six basic trigonometric functions, only cosine (and its reciprocal, secant) are even: ( ) sec ( x ) = sec x cos x = cos x However, the other four are odd: ( ) csc ( x ) = csc x tan ( x ) = tan x cot ( x ) = cot x sin x = sin x () • If f is an even function, then the graph of y = f x is symmetric about the y-axis. () • If f is an odd function, then the graph of y = f x is symmetric about the origin. (Chapter 1: Review) 1.33 PART B: DOMAINS AND RANGES OF THE SIX BASIC TRIGONOMETRIC FUNCTIONS f x () Domain Range sin x ( , ) 1, 1 cos x ( , ) 1, 1 tan x csc x sec x cot x Set-builder form: x x + n n 2 Set-builder form: ( ) { x x n ( n )} Set-builder form: x x + n n 2 Set-builder form: ( { x x n ( n )} ) ( , ) ( , 1 1, ) ( , 1 1, ) ( , ) • The unit circle approach explains the domain and range for sine and cosine, as well as the range for tangent (since any real number can be a slope). (Chapter 1: Review) 1.34 • Domain for tangent: The “X”s on the unit circle below correspond to an undefined slope. Therefore, the corresponding real numbers (the corresponding angle measures in radians) are excluded from the domain. • Domain for tangent and secant: The “X”s on the unit circle above also correspond sin to a cosine value of 0. By the Quotient Identity for tangent tan = and the cos 1 , we exclude the corresponding Reciprocal Identity for secant sec = cos radian measures from the domains of both functions. • Domain for cotangent and cosecant: The “X”s on the unit circle below correspond to a sine value of 0. By the Quotient Identity for cotangent cos 1 cot = sin and the Reciprocal Identity for cosecant csc = sin , we exclude the corresponding radian measures from the domains of both functions. • Range for cosecant and secant: We turn “inside out” the range for both sine and cosine, which is 1, 1 . • Range for cotangent: This is explained by the fact that the range for tangent is 1 . cot is 0 in , and the Reciprocal Identity for cotangent: cot = tan value tan is undefined. ( ) (Chapter 1: Review) 1.35 PART C: GRAPHS OF THE SIX BASIC TRIGONOMETRIC FUNCTIONS • The six basic trigonometric functions are periodic, so their graphs can be decomposed into cycles that repeat like wallpaper patterns. The period for tangent and cotangent is ; it is 2 for the others. • A vertical asymptote (“VA”) is a vertical line that a graph approaches in an “explosive” sense. (This idea will be made more precise in Section 2.4.) VAs on the graph of a basic trigonometric function correspond to exclusions from the domain. They are graphed as dashed lines. • Remember that the domain of a function f corresponds to the x-coordinates picked up by the graph of y = f x , and the range corresponds to the () y-coordinates. • Remember that cosine and secant are the only even functions among the six, so their graphs are symmetric about the y-axis. The other four are odd, so their graphs are symmetric about the origin. (Chapter 1: Review) 1.36 • We use the graphs of y = sin x and y = cos x (in black in the figures below) as guide graphs to help us graph y = csc x and y = sec x . Relationships between the graphs of y = csc x and y = sin x (and between the graphs of y = sec x and y = cos x ): •• The VAs on the graph of y = csc x are drawn through the x-intercepts of the graph of y = sin x . This is because csc x is undefined sin x = 0 . •• The reciprocals of 1 and 1 are themselves, so csc x and sin x take on each of those values simultaneously. This explains how their graphs intersect. •• Because sine and cosecant are reciprocal functions, we know that, between the VAs in the graph of y = csc x , they share the same sign, and one increases the other decreases. (Chapter 1: Review) 1.37 PART D: SOLVING TRIGONOMETRIC EQUATIONS Example 1 (Solving a Trigonometric Equation) ( ) Solve: 2sin 4x = 3 § Solution ( ) Isolate the sine expression. ( ) Substitution: Let = 4x . 2sin 4x = 3 3 sin 4x = 2 = sin = 3 2 We will now solve this equation for . 3 = , so will be the reference angle for our solutions 3 2 3 3 for . Since is a negative sine value, we want “coreference angles” 2 in Quadrants III and IV. of 3 Observe that sin Our solutions for are: = 4 5 + 2 n, or = + 2 n 3 3 From this point on, it is a matter of algebra. ( n ) (Chapter 1: Review) 1.38 To find our solutions for x, replace with 4x , and solve for x. 4 + 2 n, or 3 1 4 2 x= + n, or 4 3 4 4x = x= + n, 3 2 5 + 2 n 3 1 5 2 x= + n 4 3 4 ( n ) 5 + n 12 2 ( n ) 4x = or Solution set: x x = + n, or 3 2 x= x= 5 + n 12 2 ( n . § PART E: ADVANCED TRIGONOMETRIC IDENTITIES These identities may be derived according to the flowchart below. for cosine ( n ) ) (Chapter 1: Review) 1.39 GROUP 1: SUM IDENTITIES Memorize: sin (u + v ) = sinu cosv + cosu sinv Think: “Sum of the mixed-up products” (Multiplication and addition are commutative, but start with the sinu cosv term in anticipation of the Difference Identities.) cos (u + v ) = cosu cosv sin u sinv Think: “Cosines [product] – Sines [product]” tan (u + v ) = tan u + tan v 1 tan u tan v Think: " Sum " 1 Product GROUP 2: DIFFERENCE IDENTITIES Memorize: Simply take the Sum Identities above and change every sign in sight! sin (u v ) = sin u cosv cosu sinv (Make sure that the right side of your identity for sin ( u + v ) started with the sinu cosv term!) cos (u v ) = cos u cosv + sin u sinv tan (u v ) = tan u tanv 1 + tan u tan v Obtaining the Difference Identities from the Sum Identities: Replace v with (–v) and use the fact that sin and tan are odd, while cos is even. For example, sin (u v ) = sin [u + (v )] = sin u cos (v ) + cos u sin (v ) = sin u cosv cosu sinv (Chapter 1: Review) 1.40 GROUP 3a: DOUBLE-ANGLE (Think: Angle-Reducing, if u > 0) IDENTITIES Memorize: (Also be prepared to recognize and know these “right-to-left.”) sin (2u) = 2 sinu cosu Think: “Twice the product” Reading “right-to-left,” we have: 2 sinu cos u = sin (2u) (This is helpful when simplifying.) cos (2u) = cos u sin u 2 2 Think: “Cosines – Sines” (again) Reading “right-to-left,” we have: cos u sin u = cos (2u) 2 2 Contrast this with the Pythagorean Identity: cos2 u + sin2 u = 1 tan (2u) = 2 tan u 1 tan2 u (Hard to memorize; we’ll show how to obtain it.) Notice that these identities are “angle-reducing” (if u > 0) in that they allow you to go from trigonometric functions of (2u) to trigonometric functions of simply u. (Chapter 1: Review) 1.41 Obtaining the Double-Angle Identities from the Sum Identities: Take the Sum Identities, replace v with u, and simplify. sin (2u) = sin ( u + u) = sin u cos u + cosu sinu (From Sum Identity) = sin u cos u + sin u cosu (Like terms!!) = 2 sinu cosu cos (2u) = cos (u + u) = cosu cosu sin u sin u (From Sum Identity) = cos2 u sin 2 u tan (2u) = tan (u + u) tan u + tan u = (From Sum Identity) 1 tan u tan u 2 tan u = 1 tan2 u This is a “last resort” if you forget the Double-Angle Identities, but you will need to recall the Double-Angle Identities quickly! One possible exception: Since the tan (2u) identity is harder to remember, you may prefer to remember the Sum Identity for tan (u + v ) and then derive the tan (2u) identity this way. If you’re quick with algebra, you may prefer to go in reverse: memorize the Double-Angle Identities, and then guess the Sum Identities. (Chapter 1: Review) 1.42 GROUP 3b: DOUBLE-ANGLE IDENTITIES FOR cos Memorize These Three Versions of the Double-Angle Identity for cos (2u) : Let’s begin with the version we’ve already seen: Version 1: cos (2u) = cos2 u sin 2 u Also know these two, from “left-to-right,” and from “right-to-left”: Version 2: cos (2u) = 1 2 sin 2 u Version 3: cos (2u) = 2 cos2 u 1 Obtaining Versions 2 and 3 from Version 1 It’s tricky to remember Versions 2 and 3, but you can obtain them from Version 1 by 2 2 using the Pythagorean Identity sin u + cos u = 1 written in different ways. 2 2 To obtain Version 2, which contains sin u , we replace cos u with (1 sin2 u) . ( ) cos 2u = cos 2 u sin 2 u = (1 sin u ) 2 (Version 1) sin 2 u from Pythagorean Identity = 1 sin 2 u sin 2 u = 1 2 sin 2 u ( Version 2) 2 2 To obtain Version 3, which contains cos u , we replace sin u with (1 cos2 u) . ( ) cos 2u = cos 2 u sin 2 u = cos 2 u ( ) (Version 1) 1 cos 2 u from Pythagorean Identity = cos 2 u 1 + cos 2 u = 2 cos 2 u 1 ( Version 3) (Chapter 1: Review) 1.43 GROUP 4: POWER-REDUCING IDENTITIES (“PRIs”) (These are called the “Half-Angle Formulas” in some books.) Memorize: Then, sin u = 1 cos (2u) 1 1 or cos (2u) 2 2 2 cos2 u = 1 + cos (2u) or 2 2 tan2 u = sin 2 u 1 cos (2u) = cos2 u 1 + cos (2u) 1 1 + cos (2u) 2 2 2 2 Actually, you just need to memorize one of the sin u or cos u identities and then switch the visible sign to get the other. Think: “sin” is “bad” or “negative”; this is a 2 reminder that the minus sign belongs in the sin u formula. Obtaining the Power-Reducing Identities from the Double-Angle Identities for cos (2u) 2 To obtain the identity for sin u , start with Version 2 of the cos (2u) identity: cos (2u) = 1 2 sin 2 u Now, solve for sin2 u. 2 sin2 u = 1 cos (2u) sin2 u = 1 cos (2u) 2 2 To obtain the identity for cos u , start with Version 3 of the cos (2u) identity: cos (2u) = 2 cos 2 u 1 Now, switch sides and solve for cos 2 u. 2 2 cos u 1 = cos (2u) 2 cos2 u = 1 + cos (2u) cos2 u = 1 + cos (2u) 2 (Chapter 1: Review) 1.44 GROUP 5: HALF-ANGLE IDENTITIES Instead of memorizing these outright, it may be easier to derive them from the Power-Reducing Identities (PRIs). We use the substitution = 2u . (See Obtaining … below.) The Identities: 1 cos sin = ± 2 2 1 + cos cos = ± 2 2 1 cos 1 cos sin tan = ± = = 1 + cos sin 1 + cos 2 lies in. 2 Here, the ± symbols indicate incomplete knowledge; unlike when we handle the Quadratic Formula, we do not take both signs for any of the above formulas for a given . There are no ± symbols in the last two tan formulas; there is no problem there of incomplete knowledge 2 regarding signs. For a given , the choices among the ± signs depend on the Quadrant that One way to remember the last two tan formulas: Keep either the numerator or the 2 denominator of the radicand of the first formula, place sin in the other part of the fraction, and remove the radical sign and the ± symbol. (Chapter 1: Review) 1.45 Obtaining the Half-Angle Identities from the Power-Reducing Identities (PRIs): For the sin identity, we begin with the PRI: 2 sin 2 u = ( ) 1 cos 2u 2 Let u = , or = 2u . 2 1 cos sin 2 = 2 2 1 cos sin = ± 2 2 ( by the Square Root Method ) Again, the choice among the ± signs depends on the Quadrant that lies in. 2 The story is similar for the cos and the tan identities. 2 2 What about the last two formulas for tan ? The key trick is multiplication by 2 trigonometric conjugates. For example: 1 cos tan = ± 1 + cos 2 =± (1 cos ) (1 cos ) (1 + cos ) (1 cos ) (1 cos ) 2 =± =± 1 cos 2 (1 cos ) 2 sin 2 1 cos =± sin =± 1 cos sin 2 ( because a2 = a ) (Chapter 1: Review) 1.46. Now, 1 cos 0 for all real , and tan has the same 2 sign as sin (can you see why?), so … = 1 cos sin To get the third formula, use the numerator’s (instead of the denominator’s) trigonometric conjugate, 1 + cos , when multiplying into the numerator and the denominator of the radicand in the first few steps. GROUP 6: PRODUCT-TO-SUM IDENTITIES These can be verified from right-to-left using the Sum and Difference Identities. The Identities: ( ) cos (u + v ) ( ) + cos (u + v ) ( ) + sin (u v ) ( ) sin (u v ) 1 cos u v 2 1 cosu cos v = cos u v 2 1 sin u cos v = sin u + v 2 1 cosu sin v = sin u + v 2 sin u sin v = GROUP 7: SUM-TO-PRODUCT IDENTITIES These can be verified from right-to-left using the Product-To-Sum Identities. The Identities: x+ sin x + sin y = 2sin 2 x y cos 2 y x + y x y sin x sin y = 2cos sin 2 2 x + y x y cos x + cos y = 2cos cos 2 2 x + y x y cos x cos y = 2sin sin 2 2