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.
Today, we will look at our Pythagorean identities, which are a part of our trig identities. These Pythagorean identities are true statements about trig functions based on the Pythagorean theorem. You will see how they are based on the Pythagorean theorem (a2 + b2 = c2). If you look at the unit circle in trig and then look at the right triangle that is part of it, the two sides, a2 and b2, are our trig functions, while the hypotenuse, c, is the number 1.
We have a total of three identities that are part of the Pythagorean identities. We have one that involves cosine and sine, another one that involves tangent and secant, and a third that involves cotangent and cosecant.
Uses & Applications
Think of these identities as formulas that help you connect our trig functions. You can see that our trig functions are related by the Pythagorean theorem. These identities are also formulas that help us simplify our problems. You can see that if we have a function that is the sum of the square of a sine and the square of a cosine of the same angle, then we can simplify that expression as simply 1.
You will see these Pythagorean identities in use in problems that ask you to prove other trig statements as well as in simplifying more complex problems. Mathematicians also use the identities to help them figure out more complex problems using higher math, such as calculus.
Do you want to know what kinds of problems you can expect to see? Okay, let's look at a couple.
Our first problem is asking us, 'What does sec2 (70) - tan2 (70) equal?'
At first glance, we might think that we need to calculate each function, square them, and then subtract them. But if we take a moment and think about our trig identities, in particular our Pythagorean identities, we will see that there is a much simpler, quicker, and more accurate way to solve our problem. We look through our identities and we see that our second one has both the tangent and secant functions in it.
It isn't written exactly like ours, but we do see that if we use our algebra skills and rearrange the identity a bit we will see that we can get our problem function. If we subtract the tangent from both sides, our identity becomes sec2 (theta) - tan2 (theta) = 1. This identity also gives us the answer to our problem. Do you see it? It's a nice, simple 1. Wasn't that easy?
Let's look at one more problem.
Prove the trig statement sin2 (theta) = 1 - cos2 (theta).
To prove a trig statement, we start with the more complicated side, leaving the other side alone. Our mission is then to turn the more complicated side into the simpler side. In our problem, the more complicated side is the right side. So, we will work on turning the right side into the left side.
We look to our identities to see what we can substitute. We see the first identity has both the sine and cosine functions just like our problem. We also see that we can substitute the 1 in our problem with this identity. Doing that we get 1 - cos2 (theta) = sin2 (theta) + cos2 (theta) - cos2 (theta).
At this point, we now look to see what we can cancel or otherwise simplify. We see that we have a positive cosine alone with a negative cosine. These cancel each other out. What are we left with? Just the sine squared. And that's what we wanted.
So our full answer includes all of our steps. 1 - cos2 (theta) = sin2 (theta) + cos2 (theta) - cos2 (theta) = sin2 (theta). And we are done!
Let's review what we've learned now. We learned that our Pythagorean identities are true statements about trig functions based on the Pythagorean theorem. We have a total of three such identities.
These identities are used to help us solve or simplify more complicated trig problems. They are also used to help us prove other trig statements.
You'll have the ability to do the following after this lesson:
- Identify the Pythagorean identities
- Explain how to solve complicated problems or prove other trig statements with these identities
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