1. What does it mean to verify an identity?
When we verify an identity, it means that we are trying to solve the equation in such a way that when we solve for the left side of the equation, it will, in the end, equal to the right side. Verification is necessary to prove an identity to be valid. When we first solve one of these identities, we want to ignore the right side. The right side is the answer we are trying to get. We use a multitude of techniques to change the left side of the equation as to get it to the right side. We can use reciprocals, we can change the identities to another one, or we can factor out trigonometric functions. We can also, in the process, cancel anything that can be canceled out. All of these strategies are to reduce the left equation as to match the right side- the verified answer. When both sides match, then we have verified our identity.
2. What tips and tricks have you found helpful?
Tips I have found helpful is methods of solving an identity equation. If you are stuck on finding the next step in an identity, I like to see my options: can I use reciprocals? Can I use factorization? Can something be canceled out? Did I properly changed my identities? Did I try to convert everything to sin and cos? I like to use these rules to guide my problem-solving . All in all, I like to use try to have at most two trig functions when I am solving and will try to get it that way through identities. A strong tip I have is to know your identities inside-out. It is apparent that you should know your identities to solve identities. Even though you may be able to use your SSS packet to see the identities, it is much better to know them from the top of your head in order to conserve time and because if you know the identities from your memory, you will be able to think of many different paths to solve an identity, thus becoming an expert at identities.
3. Explain your thought process and steps in verifying a trig identity, in general terms.
I believe this question has been answered as I have explained it in my tips and in my answer in what it means to verify an identity. To clarify, I like to have many options at hand in solving trig identities. If the trig identity is straightforward, then of course I will take the quickest path to solving. However, when the identity consists of fractions and tan, cot, sec, etc., I prefer to conjure ways to solve them in the most simple way, through a variety of methods, as mentioned before. I find it best when I convert the original trig function to sin and cos. I then look for any times I can multiply the numerator by the reciprocal of the denominator so I can look to square trig functions. I find it more preferable to look for ways to square them because then I open the range of identities at my disposal. If that is not possible, I see if I can solve the equation by factorization. This is best done in fraction equations, as things usually cancel out. Overall, depending on the type of trig identity I have to solve, I like to use the corresponding, most efficient method to solve it. I just find factorization and reciprocal multiplication to be the two techniques I have used the most during the problems in this unit. So basically, I look at the type of problem I am doing, and I use the appropriate process, such as factorization, reciprocals, and changing identities.
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