Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
A Trigonometric Equation Identity
A trigonometric equation identity is a trigonometric equation that is held to be true. Think of these as your regular equations. They have an equals sign and are assumed to be true. Think about what it means to be an equation. Doesn't it mean that the two sides are always equal? Yes, the left side must always equal the right side. For our trigonometric equation identities, instead of numbers or variables, we will have our trigonometric functions of sine, cosine, tangent, cosecant, secant, and cotangent. So, instead of seeing 3 = 1 + 2, we will see cotangent = 1/tangent.
This video lesson is about proving or verifying these identities. How do we do that? We do this by working on just one side of the equation at a time. We work on one side of the equation and work to change it so that it looks like the other side. How do we do that? We do that by using our basic trig identities that we know for sure are true and substituting them in whenever possible.
Our Basic Identities
What are some basic trig identities that we can use to help us prove our trigonometric equation identities? These include our reciprocal identities of our trig functions.
We also have our quotient identities.
It also includes our Pythagorean identities as well as our even-odd identities.
There are other identities out there, as well. Unfortunately, there isn't room in this lesson to list them all for you. If you haven't already memorized these proven trig identities, then now is your chance to get started. You will see how easy it will make your problem-solving life.
Let's take a look at a couple of examples to see how we can use our basic identities to help us prove other trigonometric equation identities. We begin with this equation:
sin (x) csc (x) = 1
The way we work these problems is we start with the more complicated side to see if we can simplify it so that it looks like the other, simpler side. For this problem, the more complicated side is the left-hand side. So we will start there. Remember, since we are trying to prove or verify this equation identity, we work only with the one side. We leave the other side completely alone. Our goal is to reach the other side, so we never touch it. Let's begin.
We look at our left-hand side and think, what can we do help us simplify this? The easiest way to begin is to rewrite everything in terms of cosine and sine. We have a sine function, great! We have a cosecant function. Okay. We can rewrite that in terms of the sine function, since we know the identity of the cosecant function is the reciprocal of the sine function. So, now we have this:
Now, how can we simplify this further? We see we have a sine divided by a sine. Well, can't we cancel them out? Yes, we can. Doing that, what do we get? We get 1. Woohoo! We did it!
Our answer then is our whole string of work. Each step is part of the answer. What you see here is our complete answer.
Let's try another one.
cot (x) sin (x) = cos (x)
The left-hand side is more complicated, so we will work on that side. Again, we rewrite everything in terms of cosine and sine. We see a cotangent function, which we know is the same as cosine over sine. So we have this:
Hey, we have a sine in the numerator and denominator! We can cancel those out. What are we left with? Cosine! We did it again! So our complete answer is this:
As you've seen, all we are doing is substituting in our known basic identities. The hardest part is now knowing how things will cancel or simplify. But as you've seen, as soon as you start working on it, the problem somehow starts working itself out for you. Sometimes, it does take some trial and error. But you will eventually find your answer.
Let's review what we've learned now. We learned that a trigonometric equation identity is a trigonometric equation that is held to be true. This video lesson taught us how to prove or verify these equation identities. The first step for proving our trigonometric equation identities is to start with the more complicated side to work on. Then we rewrite everything in terms of cosine and sine to see what we can simplify. Then we keep working and substituting in our basic trig identities until we reach our other side. Our full answer includes all the steps we took to get our one side to match the other side.
Measure your ability to do the following things when the lesson concludes:
- Define trigonometric equation identity
- Identify the basic trig identities
- Prove or verify trig equation identities
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