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Identities zAn identity is a statement that remains true, regardless of the values of the variables that appear in it. {For example: x+x=2x — always true. zA statement which is true for only some values of the variables is a conditional equality. {For example: 2x+5=13 — only true for x=4. 1 Trigonometric Identities zA trigonometric identity is an identity involving a combination of trigonometric functions. zWe have looked at some identities before: the definitions of tangent, cosecant, secant, and cotangent in terms of sine and cosine. 2 Quotient and Reciprocal Identities zQuotient Identities sin θ tan θ = cos θ cos θ cot θ = sin θ zReciprocal Identities 1 sec θ = cos θ 1 csc θ = sin θ 1 cot θ = tan θ 3 Trigonometric Identities zThese identities, and others we will learn, are useful for simplifying trigonometric expressions and equations. zProving that, in an equation, one side is always* equal to the other is known as establishing an identity. * here, “always” means “in the domains of the functions involved” 4 Establishing Identities zEstablishing an identity is just like solving a fun mathematical jigsaw puzzle! {One side of the equation consists of the “pieces” you have to work with. {The known identities are what you use to manipulate the pieces to fit together. {The other side of the equation is the picture on the box… that is your goal! 5 Example #1 zWe want to show that sin x cot x = cos x zWe should start on the left side, since it is more complicated, and make it look like the right. zWe can algebraically manipulate either side of the equation, but can’t move anything across the equal sign! 6 Example #1 zWe know the definition of cot x in terms of sin x and cos x: cos x cot x = sin x zConverting to sines and cosines is a good idea, because, chances are, things will cancel very nicely. 7 Example #1 zTherefore, sin x cot x cos x sin x · sin x cos x = cos x = cos x = cos x ¤ zThe identity has been established. 8 Examples #2-3 csc x tan x sin x 1 · sin x cos x 1 cos x sec x = sec x = sec x = sec x = sec x ¤ sin2 θ sec θ csc θ 1 1 · sin θ · sin θ · cos θ sin θ sin θ cos θ tan θ The notation sin2 θ means (sin θ)2 . = tan θ = tan θ = tan θ = tan θ ¤ 9 Pythagorean Identities zOn the unit circle, the sine and cosine of the angle describe two legs of a right triangle, while the radius (of 1) is the hypotenuse. zThus, it follows from the Pythagorean theorem that 2 2 sin θ + cos θ = 1 10 Pythagorean Identities zUsing this identity, we can create the other two Pythagorean identities. (Do you see how?) 2 2 sin θ + cos θ 2 tan θ + 1 2 1 + cot θ = = = 1 2 sec θ 2 csc θ 11 Example #4 zWe want to show that 2 2 2 cos β − sin β = 1 − 2 sin β zLet’s start on the left side. We can rewrite the first Pythagorean identity in terms of one of the given terms to eliminate it. sin2 β + cos2 β = 1 ⇒ cos2 β = 1 − sin2 β zAfter we do that, we can combine like terms and simplify. 12 Example #4 zTherefore, cos2 β − sin2 β ¢ ¡ 2 1 − sin β − sin2 β 2 1 − 2 sin β = = = 1 − 2 sin2 β 1 − 2 sin2 β 2 1 − 2 sin β ¤ zWe could have eliminated the other term on the left or either term on the right. All methods would establish the identity. 13 Example #5 zWe want to show that 2 (1 + cos x)(1 − cos x) = sin x zTo establish this identity, we can FOIL the left side and then cancel terms. (1 + cos x)(1 − cos x) 1 − cos x + cos x − cos2 x 1 − cos2 x 2 sin x = = = = sin2 x sin2 x sin2 x 2 sin x ¤ 14 Example #6 zWe want to show that cot x + tan x = csc x sec x zFirst, let’s rewrite everything in terms of sines and cosines. cos x sin x 1 1 + = · sin x cos x sin x cos x zTo add, we need a common denominator, which in this case is sin x cos x . 15 Example #6 zTherefore, sin x cos x + sin x ¶ cos x µ ³ cos x ´ cos x sin x sin x + sin x cos x cos x sin x cos2 x sin2 x + sin x cos x sin x cos x cos2 x + sin2 x sin x cos x 1 sin x cos x = = = = = 1 1 · sin x cos x 1 sin x cos x 1 sin x cos x 1 sin x cos x 1 sin x cos x ¤ 16 Example #7 zWe want to show that sin θ 1 − cos θ = sin θ 1 + cos θ zTo establish this identity, we take the more complicated right side (since it has addition in the denominator) and multiply by the conjugate of the denominator. zRemember, you cannot cross multiply! 17 Example #7 zThe conjugate of a binomial takes the original two factors and reverses the sign in between. The conjugate of a+b is a–b. zIn this example, the conjugate of 1 + cos θ is 1 − cos θ . zMultiplying the right side by a fancy form of 1 involving this conjugate will create something that FOILS and cancels nicely. 18 Example #7 zTherefore, 1 − cos θ sin θ 1 − cos θ sin θ 1 − cos θ sin θ 1 − cos θ sin θ 1 − cos θ sin θ = = = = = sin θ 1 + cos θ sin θ 1 − cos θ · 1 + cos θ 1 − cos θ sin θ(1 − cos θ) 1 − cos2 θ sin θ(1 − cos θ) sin2 θ 1 − cos θ sin θ 19