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Unformatted text preview: (Section 5.2: Verifying Trig Identities) 5.10 SECTION 5.2: VERIFYING TRIG IDENTITIES PART A: EXAMPLE; STRATEGIES AND “SHOWING WORK” One Example; Three Solutions Verify the identity: csc θ + cot θ tan θ + sin θ = cot θ csc θ . Strategies and “Showing Work” To verify an identity like this one, use the Fundamental Identities and algebraic techniques to simplify the side with the more complicated expression stepbystep until we end up with the expression on the other side. You may think of this as a simplification problem where the “answer” is given to you. The “answer” may be thought of as the top of a jigsaw puzzle box, the TARGET that you are aiming for. This is a strategy to keep in mind as you perform your manipulations. Warning: Instructors generally want their students to “show work.” In the simplification or verification process, you should probably write a new expression every time you apply a Fundamental Identity and every time you execute a “major” algebraic step (this may be a matter of judgment). If you are applying Fundamental Identities to different pieces of an expression, you may be able to apply them simultaneously in one step, provided that it is clear how and where they are being applied. In this class, you will typically not be required to write the names of the various identity types you are using, but they will often be written in solutions for your reference. The lefthand side (LHS) seems more complicated in this example, so we will operate on it until we obtain the righthand side (RHS). In principle, you could begin with the RHS, or you could even work on both sides simultaneously until you “meet” somewhere in the middle. Some instructors may object to the latter method, however, perhaps because it may seem “sloppy.” Even then, it could still inspire a more linear approach. There are often different “good” approaches to problems such as these. You don’t necessarily have to agree with your book’s solutions manual! (Section 5.2: Verifying Trig Identities) 5.11 Solution (Method 1) (This may be the least efficient approach, though.) Remember, we want to verify: csc θ + cot θ tan θ + sin θ = cot θ csc θ csc θ + cot θ tan θ + sin θ = 1 sin θ + cos θ sin θ sin θ cos θ + sin θ Reciprocal and Quotient Identities ( ) We are breaking things down into expressions involving sin θ and cos θ . They are like common currencies. We can begin by simplifying the numerator (“N”) and the denominator (“D”) individually. Tip: It may help to express sin θ as sin θ 1 . = 1 + cos θ sin θ sin θ cos θ + sin θ 1 ← We already had a common denominator....
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 Fall '11
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 Calculus, Fractions, Sin, Cos, Elementary arithmetic, Mathematics in medieval Islam

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