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Prior to Calculus: Trigonometric Identities

Hey, guys. Trigonometric identities never go away. Ever. If you need to do anything with math, this is sure to come up at some point. They are so important to learn and use properly! There are many different ones, so I couldn’t possibly list and teach all of them to you. There is a whole course devoted to trig, that many people take in highschool. I’ll show you the ones that I use most often, and all the ones that I can remember off the top of my head. I’m only going to show you the ones I can remember without looking them up, because that is obviously the ones that are most useful to me. If I need to know it, I would have remembered it by now.

So, trigonometry isn’t just about triangles. It’s about the relationship between the angles and lengths of the sides. We describe these relationships using trig functions. There are six of them, and you should become comfortable with them all. You can do some really cool things with them. They come up in derivatives, integrals, and can be used to simplify answers in many ways.

Reciprocal Identities

These ones are fairly easy to remember, as there is a symmetry about them. Here’s my trick: List out the 6 trig functions. The two on the outside go together, the second one and the second to last one goes together, and the two middle ones go together. Let me show you how I picture it. Write them out vertically, and draw lines to match them up. This will tell you that sin goes with csc, cos goes with sec, and tan goes with cot.

See? If you are ever in doubt, just write this out very quickly. You’ll never get it wrong!

Tangent and Cotangent

These two identities are extremely important, and not very hard to remember. Keep them in mind, because you will definitely need them.

Other Important Identites

 

These three are also very important to know. The first one comes up most often, and the third one the least often. However, I’ve seen them all come up often! Remember that these last three use the squares of trigonometric functions. Try not to mess that up.

How are they used?

I’m going to teach you the technique I use to simplify complex trigonometric functions. You see these often throughout calculus. I’m not going to show you specific problems, just big expressions that I’ll simplify. You need to be on the lookout. If you have an expression with many different trig functions inside it, try to think if it can be simplified. Follow my method, and see what you can do. Sometimes you’ll be surprised, and things can simplify dramatically.

Step 1. Look for obvious simplifications. That should be your first step all the time. If you can blatantly see one of the identities, simplify it right away. Don’t skip over them just to follow my process. I tend to have a gut feeling about simplifying expressions, because I have a lot of experience with it. If you feel like something might work, go with that gut feeling. It’s good to explore these identites and try out things on your own.

Step 2. Convert everything to sin’s and cos’s. Do this using reciprocal identities and tangent and cotangent identities. Now, you can see if you can combine like terms or eliminate fractions.

Step 3. Now, do all the cancelling that you can! Use factoring and the last three identities to simplify the expression.

Step 4. The last step is to convert back to the most proper expressions. For example, if you have a sin in the denominator, move it to the numerator, making it a csc. Convert back to tangents and cotangents when necessary, and other things like that. Get rid of as many fractions as you can. That’s how you make it look clean. That’s what the final answer should look like.

Here are two examples of complex expressions, that I’ll simplify as much as possible. There are many other things that happen, but I can’t show every type of example. There are too many. Following this process with help you simplify just about any trigonometric expression, which is necessary in very many situations!

Immediately, I don’t see any easy simplifications. So, I convert everything to sin’s and cos’s. Now, I see that the numerator can be turned into just a sin squared, because of one of the identities. Also, the cos is in the denominator of the denominator, and we can fix that. When you divide by a fraction, it is the same as multiplying by the reciprocal. So, if you flip the denominator, and multiply by it, the cos moves to the numerator, and the sin squared stays in the denominator. Then, there is a very easy cancellation of the sin squareds, and you are left with just a cos!

In this expression, something sticks out to me. I’m always on the lookout for seeing a sin squared and a cos squared near each other. If they are being added together, you can cancel them out using a trig identity. They become just a 1, and it is so nice to simplify a large expression into something so small! So, the first thing I did was factor the cosecant, to get the sin squared plus cos squared on its own. Then, I cancel them, using a trig identity. The next thing I did was make everything a sin or cos. After that, I see one more cancellation. Then, I converted the cos in the denominator to a sec, just to make it a little cleaner looking. I hope you could follow me!

I hope this has helped you! As always, leave a comment below or email me at admin@calculustricks.com with any questions or comments.

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  1. Rob says:

    Just saved me for my upcoming calc midterm!

  2. frbry says:

    this really helped me, thanks a lot.

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