M1410504Handout

M1410504Handout - HANDOUT ON SECTIONS 5.4 AND 5.5 MORE TRIG...

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HANDOUT ON SECTIONS 5.4 AND 5.5: MORE TRIG IDENTITIES – MEMORIZE! SUM IDENTITIES Memorize: sin u + v ( ) = sin u cos v + cos u sin v Think: “Sum of the mixed-up products” (Multiplication and addition are commutative, but start with the sin u cos v term in anticipation of the Difference Identities.) cos u + v ( ) = cos u cos v sin u sin v Think: “Cosines [product] – Sines [product]” tan u + v ( ) = tan u + tan v 1 tan u tan v Think: " Sum 1 Product " DIFFERENCE IDENTITIES Memorize: Simply take the Sum Identities above and change every sign in sight! sin u v ( ) = sin u cos v cos u sin v (Make sure that the right side of your identity for sin u + v ( ) started with the sin u cos v term!) cos u v ( ) = cos u cos v + sin u sin v tan u v ( ) = tan u tan v 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 cos v cos u sin v
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DOUBLE-ANGLE (Think: Angle-Reducing, if u > 0) IDENTITIES Memorize: (Also be prepared to recognize and know these “right-to-left”) sin 2 u ( ) = 2 sin u cos u Think: “Twice the product” Reading “right-to-left,” we have: 2 sin u cos u = sin 2 u ( ) (This is helpful when simplifying.) cos 2 u ( ) = cos 2 u sin 2 u Think: “Cosines – Sines” (again) Reading “right-to-left,” we have: cos 2 u sin 2 u = cos 2 u ( ) Contrast this with the Pythagorean Identity: cos 2 u + sin 2 u = 1 tan 2 u ( ) = 2 tan u 1 tan 2 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 trig functions of (2 u ) to trig functions of simply u .
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Obtaining the Double-Angle Identities from the Sum Identities: Take the Sum Identities, replace
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M1410504Handout - HANDOUT ON SECTIONS 5.4 AND 5.5 MORE TRIG...

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