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Module 6 Lesson 4 Using Trigonometric Identities – Part 3 Simplifying Trigonometric Expressions Trigonometric Identities are found in many different areas of mathematics and science. In addition to what we’ve been doing, there are two more main uses of trig identities. The first use is to simplify trigonometric expressions. Consider that there’s a science experiment in progress. The scientist has found an expression to go along with some data that he’s collected. The expression looks like this: sec cos2 cot Looks a little complicated…and time consuming. If only there were some way to write that in simpler terms. Fortunately for us, there is. We’ve talked about Pythagorean, Reciprocal and Quotient Identities, among others. We can make substitutions using these identities which will make good things happen for us. Remember this Reciprocal Identity? sec 1 cos Remember this Quotient Identity? cot cos sin Let’s substitute those into our original expression: sec cos2 cot 1 cos2 becomes cos cos sin We can now use some basic Algebra to simplify. Notice on the top that one of the cosines cancels out: 1 1 cos2 cos cos sin Now we divide cos by becomes cos cos sin cos . To do this, we multiply by the reciprocal: sin cos cos sin becomes cos sin cos The cosines cancel and we’re left with sin . What did we just show? We showed that: sec cos2 sin cot Now if you were the scientist doing this experiment, which would you rather write out in your notes hundreds of times? You can double check your result by plugging some different values in for theta. After all, if these two things are actually equal, you should be able to plug in any value at all for theta and get a valid equation. So select a random number. I’m going to choose 16 degrees. Evaluate this: sec(16) cos2 (16) cot(16) Is it the same as: sin(16) It is. (If you don’t think it is, you typed something in incorrectly). How do you know what to do? Sure this simplification is simple if your instructor tells you how to do them. But how do you make some of these decisions for yourself? Although I can’t anticipate and help you solve every problem, I can provide some guidelines and tips for you to keep in mind as you simplify these trigonometric expressions. These tips aren’t in any order. Sometimes a tip will help in one problem and make 2 things worse in another. These are just some different suggestions that are intended to give you a starting point to simplify. TIP #1: Try to write everything in terms of sine and cosine The reason behind this is simple. TAN, COT, CSC, and SEC can all be written in terms of sine and cosine. If everything in the problem says SIN or COS, you might get some things to cancel out. In fact, that’s what happened in our example. TIP #2: If you see a ‘1’ or a squared term – think Pythagorean Identities Countless times in these identities, you’ll run into a substitution involving one of our 3 Pythagorean Identities or some variation thereof. You should be aware of the 3 main identities and also how you might cleverly rewrite them. For example, you know our first Pythagorean Identity is: sin 2 cos2 1 But do you also realize you could write that as either of these: cos2 1 sin 2 sin 2 1 cos2 Or even possibly: cos2 sin 2 1 sin 2 cos2 1 Be aware. SEC and TAN go together. CSC and COT go together. TIP #3: Use Basic Algebra! There will be many times when you add/subtract fractions, factor, multiply/divide (FOIL!), find a common denominator, combining like terms, squaring, etc. etc. Keep an open mind. TIP #4: There’s probably more than one way to simplify There are going to be problems that will have more than one way to simplify. There’s no ‘right’ or ‘wrong’ way to simplify, as long as at the end, you have the simplest form. TIP #5: The time factor 3 If you’re spending more than 5-10 minutes on any one problem, you’re probably either doing it wrong, or you just don’t know how to do it. Don’t be afraid to stop and ask. These problems aren’t intended to be terrible and timeconsuming. There’s no reason to spend more than 5-10 minutes on a problem. It just causes bad feelings and unnecessary frustration. The best way to explain these problems is actually just to illustrate through some examples. I’ll try to include as much explanation as I can, although I’ll start abbreviating steps after a point. Example 1: Simplify: sin 2 sec 2 1 Here, we see a ‘1’ and a trig function squared. This screams Pythagorean Identity. Locate the identity that involves secant and 1: 1 tan 2 sec 2 Doing a little shuffling gives us: tan 2 sec 2 1 We can now make the substitution in the original expression: We rewrite tangent in terms of sine and cosine: sin 2 tan 2 sin 2 sin 2 cos2 sin 2 sin 2 cos2 2 cos2 Use basic algebra to simplify: 2 sin 1 sin 2 cos And we’re done. We’ve simplified sin 2 into cos2 2 sec 1 Example 2: sin cot cos Simplify: Here, we’ll start by writing cotangent in terms of sine and cosine: sin cos cos sin 4 Some basic algebra gives us: sin cos2 1 sin You can see that I’ve rewritten sine as a fraction over 1. This is because I see that am supposed to be adding these two parts together. We do that with a common denominator. In this case, the common denominator is sin . So I’ll need to multiply sin the first fraction by : sin sin cos2 1 sin sin sin cos2 ( ) sin 1 sin 2 2 sin cos sin sin Adding the fractions gives: sin 2 cos2 sin Another convenient Pythagorean Identity Substitution: which simplifies to: csc So our end result: sin cot cos simplifies to csc Example 3: Simplify: 1 sin (sec x tan x )(1 sin x ) Start with some basic algebra (specifically FOIL): (sec x tan x)(1 sin x) sec x sec x sin x tan x tan x sin x Put everything in terms of sine and cosine: sec x sec x sin x tan x tan x sin x 1 1 sin x sin x ( )(sin x ) ( )(sin x ) cos x cos x cos x cos x Order of operations tells us to multiply: 5 1 1 sin x sin x ( )(sin x ) ( )(sin x ) cos x cos x cos x cos x 1 sin x sin x sin 2 x ( ) cos x cos x cos x cos x We now have a common denominator, so let’s combine all the fractions. Conveniently, the ‘sin x’ terms cancel out: 1 sin x sin x sin 2 x ( ) cos x cos x cos x cos x 1 sin 2 x cos x 1 sin 2 x cos2 x cos x cos x cos x Pythagorean and simplify: So we just showed that: (sec x tan x )(1 sin x ) simplifies to cos x Verifying Trigonometric Identities: What we’ve done so far is to look at expressions and simplify them. The other main part of this process is verifying trig identities. This is what happens when instead of an expression, you’re presented with an equation: cos x(tan 2 x 1) sec x To verify this identity, we need to show that the two sides of the equation are equal. We do that in the same way we did the problems above. We try to simplify and manipulate the left side until it looks exactly like the right side. When we verify identities, we generally try to change one side only during the process. It might be tempting to involve the other side in some fashion, but try not to. Try to exclusively work on one side (either left or right). Will it always be possible to only work with one side? Probably not. But try. cos x(tan 2 x 1) sec x This example illustrates one other point. How would you start this problem? There are two distinct ways – either of which will eventually lead to the correct answer. Some of 6 you might have started by using a Pythagorean Identity substitution. Some of you might have thought “Let’s get rid of the parenthesis using the distributive property”. Both are valid. Both will work. This just goes to show you that there will be several ways to verify these identities. There isn’t usually a “right” and “wrong” way. There might be a way that’s quicker, but as long as you verify what you’re supposed to, it shouldn’t matter how you get there. Here are the solutions. I think you’ll agree that both work, but one is quicker. I’ll leave the reasoning to you. Ask if you don’t understand. Example 4: Verify this trigonometric identity: cos x(tan 2 x 1) sec x cos x(tan 2 x 1) sec x Here is the first method: cos x(sec 2 x ) sec x 1 cos x( 2 ) sec x cos x 1 sec x cos x sec x sec x Nice and quick. Here is the other way. Wow. 7 cos x (tan 2 x 1) sec x (cos x )(tan 2 x ) cos x sec x sin 2 x (cos x )( 2 ) cos x sec x cos x (cos x )(sin 2 x ) cos x sec x cos2 x 1 (cos x )(sin 2 x ) cos x cos 2 x ( ) sec x cos2 x 1 cos2 x (cos x )(sin 2 x ) cos3 x sec x cos2 x cos2 x (cos x )(sin 2 x ) cos3 x sec x cos2 x sin 2 x cos2 x sec x cos x 1 sec x cos x sec x sec x Let’s do one more. Example 5: Verify this trigonometric identity: cot 2 x 1 sin 2 x 2 csc x Start with some Pythagorean action: Sines and cosines: cos2 x sin 2 x 1 sin 2 x 1 sin 2 x Using Algebra on the left side, we can see: So now we have: cot 2 x 1 sin 2 x 2 1 cot x cos2 x 2 2 sin 2 x ( cos x )( sin x ) cos2 x 1 sin 2 x 1 2 sin x cos2 x 1 sin 2 x 8 One final Pythagorean substitution: 1 sin 2 x 1 sin 2 x Keep in mind that there’s no set way to do these processes. There are guidelines and tips to help you, but there’s no set way to simplify/verify. There might be different ways to solve problems, some quicker, some slower. Be patient and persistent with these problems. As you do more and more of them, they’ll get easier. 9