In this lesson, I'm going to cover some of the basic formulas that you'll see if you're just starting out in trigonometry. So here's the first one. Let's say we have a circle.
And let's say this is the radius of the circle, as well as this. So this point here is the center of the circle. So that's the center. This is the radius, r.
The radius is the distance between the center of the circle and any point on the circle. The arc length represented by the symbol s will be this distance here and this is the angle theta. The equation that relates arc length to the angle theta is this.
The arc length s is equal to theta times r. In this equation theta is in radians, not degrees. r is the radius. Now, if r is in feet, the arc length will be the same.
It's going to be in the same unit, feet. If the radius is in meters, then the arc length will be in meters. Keep in mind, 360 degrees is equal to 2 pi radians.
So you could use that to convert between radians and degrees. If you want to know how to do that, I have a video on YouTube that explains how you can convert from radians to degrees and vice versa. Just type in radians to degrees organic chemistry tutor in the YouTube search bar and you'll see that video.
Now let's say if you want to calculate the area of a sector of a circle as opposed to the area of the entire circle. There's two formulas that you can use to do that. The first one is this one. The area is equal to one half theta times r squared. In this formula, theta is in radians.
The other one is the area is equal to theta divided by 360 degrees times pi r squared. In both cases, r represents the radius of the circle but in this equation, theta is measured in degrees as opposed to in radians. Now let's talk about right triangle trigonometry. Let's discuss the formulas associated with it.
So let's say we have a right angle of 90 degrees, and here we have the angle theta. This side is opposite to the angle, this side is adjacent to it, and across the right angle is the hypotenuse, the longest side of the triangle. Now, it's good to know this expression, SOHCAHTOA, because it helps you to remember the formulas of three trig ratios, sine, cosine, and tangent. So the first part, the SOH, tells you that sine theta is equal to the opposite side divided by the hypotenuse.
So that's the first trig ratio you want to be familiar with. Ka tells you that cosine of the angle theta is equal to the adjacent side of the triangle divided by the hypotenuse. And Toa tells you that tangent theta is equal to the opposite side over the adjacent side.
Now, there's something called the reciprocal identities. It will give you the other three trig functions. So the first one, secant, is the reciprocal of cosine. Secant is 1 over cosine. So remember how cosine was adjacent over hypotenuse?
Secant is the reciprocal of that formula. It's hypotenuse over the adjacent side. Cosecant theta is the reciprocal of sine.
So recall that sine theta is opposite over hypotenuse, which means that cosecant theta is going to be the reciprocal of that. It's hypotenuse over the opposite side of the right triangle. Next we have cotangent, which is the reciprocal of the tangent function. So tangent was opposite over adjacent. Cotangent is going to be adjacent over the opposite side.
By the way, for those of you who want to print out of these formulas, I do have a formula sheet. I'm going to post a link to that formula sheet in the description section below this video. Now, that formula sheet is going to have other formulas that I'm not going to cover in this video because there's a lot of trig formulas. So if you want a list of all those other formulas, feel free to check out that formula sheet.
Now let's move on to the quotient identities. Tangent theta is equal to sine theta divided by cosine theta. Cotangent theta, which is the reciprocal of tangent, is going to be the opposite of that.
It's cosine divided by sine. So those are the quotient identities. Next, we have the Pythagorean identities. Think of the formula a squared plus b squared equals c squared.
We have sine squared plus cosine squared is equal to 1. And there's other variations of that. There's 1 plus tangent theta, tangent square theta is equal to secant square theta. And a similar one is 1 plus cotangent square theta is equal to cosecant square theta.
So those are the Pythagorean identities. Next up are the even-odd identities. sine theta is sine negative theta is negative sine theta so that's considered an odd function the reason for that notice the sign change from positive to negative tangent of negative theta is equal to negative tan theta Cosecant negative theta is equal to negative cosecant theta. And cotangent of negative theta is negative cotangent theta. Those are the odd functions.
The even functions are the ones associated with only cosine. They don't have sine in it. All the ones that do have sine are the odd functions. Tangent is sine over cosine.
Cosecant is one over sine. Cotangent is cosine over sine. So if there's any related sine functions, it's going to be odd. Even is only cosine.
Cosecant is one over sine. Cosine negative theta is equal to positive cosine. The reason why it's even, notice that the signs on the outside remain the same.
They don't change. Secant negative theta is the reciprocal of cosine. So we should expect that to be an even function as well, which it is.
So those are the even odd identities. Next up are the co-function identities. Cosine, which is the co-function of sine, it differs by sine relative to, by 90 degrees, so to speak. So an example of this would be cosine of 30 degrees is equal to sine of 60. 30 and 60 are complementary angles. They add up to 90. Cosine of 20 degrees is equal to sine of 70 degrees.
Cosine of 10 degrees is equal to sine of 80 degrees. So when the two angles add up to 90, when they're complementary, cosine and sine will have the same value. sine 90 minus theta is equal to cosine theta tangent 90 minus theta is equal to its co function cotangent cotangent 90 minus theta is equal to tangent theta secant 90 minus theta is equal to cosecant theta. And cosecant, 90 minus theta is equal to secant theta.
So those are the co-function identities. Next, let's talk about the double angle formulas. sine 2 theta is equal to 2 sine theta cosine theta That's the most common double angle formula for sine.
A less common version is this one sine 2 theta is a loud train outside my house. Sine 2 theta is 2 tangent theta divided by 1 plus tangent squared theta. That's another double angle formula for sine, but it's not very common.
But it's helpful to have it just in case you need it. Now, The double angle formula for cosine 2 theta is this. There's different formats.
There's cosine squared minus sine squared. It's also equal to 2 cosine squared theta minus 1. And it's equal to... 1 minus 2 sine squared theta. Those are the three common forms of cosine 2 theta.
Now there is a form that involves tangent. This is less common, but here it is. It's 1 minus tan squared theta over 1 plus tan squared theta.
Now the double angle formula for tangent 2 theta is this. It's 2 tangent theta over 1 minus tan squared theta. Notice that it's very similar to sine 2 theta.
The difference is for sine, instead of a minus on the bottom, you have a plus for sine 2 theta. So far we've covered the trig ratios, the reciprocal identities, the quotient identities, the Pythagorean identities, the even odd identities, co-function identities, and the double angle formulas. There's also the half angle formulas, the triple angle formulas, the power reducing formulas.
There's the sum and difference identities, polar equations, sum to product formulas, product to sum formulas, law of sines, law of cosines, law of tangents, the area of the triangle, Heron's formula, and other stuff too. I'm going to put the rest of those formulas in the formula sheet. So for those of you who want the rest of those formulas and identities, feel free to take a look at that in the description section below. Thanks for watching.