JAMES SOUSA: Welcome to a video that will demonstrate how to find trigonometric function values on the graphing calculator. Calculators are able to determine trigonometric function values in degrees and radians. However, most calculators cannot return the values in radical form unless it has the computer algebra system. Most return decimal approximations unless the values are rational. 

Many years ago, trigonometric values were determined by tables, but this is very rare today. This video uses the TI-84 graphing calculator, though most graphing calculators are similar. This video will only use angle measures in degrees. Radians will be covered later, though the process is essentially the same. So the first thing we need to do is make sure that our calculator is in degree mode. So if we press the Mode key, the third row has the option of radians or degrees. 

So if we press down to the third row, right arrow once over to Degrees, we need to press Enter. And now it's in degree mode. So if we press 2nd Mode, which is the quit function, we're back at the home screen. And we're ready to go. So if they want to find the sine of 30 degrees, we simply press the Sine key, 30, close the parentheses, and press Enter. And we see that it's 0.5, which, of course, is equivalent to 1/2. 

If we want the cosine of 45 degrees, press the Cosine key, 45, close it, press Enter. And you can see here that this is an irrational value. It's actually the square root 2 over 2. But in decimal form, it's approximately 0.7071. 

And the same procedure for the tangent of negative 264 degrees-- there is a Tangent key. So we press Tangent, negative 264. Press Enter. And we can see that it's approximately negative 9.5144. 

This seems like a very straightforward process. However, when we consider the function values of cosecant, secant, and cotangent, we do need to be a little more careful. Remembering the relationships among the trigonometric functions is important. So, for example, if we want to find the secant of 102.5 degrees, there is no secant key on the calculator. 

However, there's a couple of ways of doing this. Remember that secant theta is equal to 1 over cosine theta, or they are reciprocals of one another. So using this relationship, we could find the secant of 102.5 degrees as 1 divided by cosine 102.5 degrees. So notice we close the parentheses here. And then we also need to close it again for the denominator. Press Enter. And there's our approximate value, negative 4.6202. 

Now, we could have obtained the same value by finding the cosine of 102.5 degrees and then taking the reciprocal. Let's just go ahead and show that. So if I found the cosine of 102.5, pressed Enter, if I want to find the reciprocal of this, I can simply press the x to the negative power. And it will convert this to negative 4.6202, as we had above. 

One thing we can't use is, we can't use this cosine which looks like to the negative 1 power. That's actually the inverse cosine function or arccosine function, which we'll cover later. So again, if we want to find the cosecant of 432 degrees-- and because there's no Cosecant key, we'll find 1 over the sine of this angle, so 1 divided by the sine of 432 degrees. 

And we have our approximate value, 1.0515. Or again, I'll show this one more time. If I found the sine of 432 degrees and then take the reciprocal of this, the result would be equivalent, as we see here. 

And lastly, we come to the cotangent of negative 23.45 degrees. Remember that cotangent theta is equal to 1 over tangent. This would work. Or we could also use cosine theta divided by sine theta. I'm going to go ahead and use 1 divided by tangent theta to find the cotangent of negative 23.45 degrees. And remember, you have to use the Negative key. You can't use the Minus key for the negative or you'll get an error. Press Enter. And there's our approximate value. 

That's pretty much it for finding trigonometric function values on the graphing calculator. And I will mention now that in another video, I will discuss how to find a possible angle if you are given the trigonometric function value. And that has to deal with inverse trigonometric functions, as we see here. The inverse sine function is written like this or arcsine x. Inverse cosine function is written like this or arccosine x-- and similarly for the inverse tangent function. 

So that's what these three keys in blue are involved with. I hope you found this video helpful. Thank you, and have a good day.