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The Tangent Function - Problem 2
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Let's do a quick review of the unit circle definitions of sine, cosine and tangent. Here is my unit circle, my angle theta is drawn in standard position, here is the point p where the terminal side crosses the edge of the circle.
We defined x, the x coordinate to be cosine of the theta, y to be sine of the theta and we defined tangent theta as y over x. Now I want to show you an identity, because tangent theta equals y over x and y is sine theta, this is the same as sine theta over x which is cosine theta. Tangent theta equals sine theta over cosine theta, that's an identity for tangent and a lot of people use that as its definition which is fine.
Another identity comes from the fact that the unit circle has the equation x² plus y² equals 1, because x is cosine theta and y is sine theta this becomes a very important identity cosine squared theta plus sine squared theta equals 1. This is called the Pythagorean identity and we'll be using this a lot in the near future.
So we have two big identities; the tangent identity and the Pythagorean identity and before we go on I want to take a quick look at Geometry Sketchpad because I want to review what we learned about tangent and the other sine functions on the unit circle so let's take a look.
So here we're on Geometry Sketchpad, this is the unit circle and remember the unit circle is how we define the sine, cosine and tangent of an angle. My angle is drawn here along the arc of the circle and this vertical length here gives the sine of the angle, this horizontal length gives me the cosine.
Now in the first quadrant both the sine and cosine are positive as you can see here, however if I move to the second quadrant you can see that the cosine becomes negative not because this directed distance is now negative, I mean the second quadrant. In the third quadrant, both sine and cosine are negative and in the fourth quadrant only sine's negative.
So how does this affect tangent? Well tangent is defined as the sine over cosine, so when both of those are positive, the tangent will be positive and they're both positive in the first quadrant. In the second quadrant cosine is negative so you're going to have a positive over a negative, you're going to get negative values. The tangent is negative in the second quadrant.
In the third quadrant both sine and cosine are negative, so the tangent is going to be positive as you can see up here, 1.05 and in the fourth quadrant sine is negative but cosine is positive, we have a negative over a positive and that is going to give us a negative answer.
So just to review, tangent is going to be positive in the first and third quadrants because that's where sine and cosine are, they're both positive or both negative. Tangent oops let me move that back. Tangent will be negative in the second quadrant because cosines are negative and sine is positive and tangent will be negative over here in the fourth quadrant because cosine is positive and sine is negative. Basically in a quadrant where sine is positive and cosine is negative or cosine is positive and sine is negative, the tangent will be negative.
Let's review what we've learned. Tangent is positive here and here in the first and third quadrants. Sine is positive here and here and cosine is positive here and here. How do we remember all that? Well there's a mnemonic device. Remember that in the first quadrant all three of those functions are going to be positive. In the second quadrant, sine is going to be positive. In the third quadrant tangent is going to be positive and in the fourth quadrant the cosine.
So first quadrant A, second quadrant S, third quadrant T, fourth quadrant C. The mnemonic for this is 'All Students Take Calculus', I know it sounds a little like propaganda, but if that helps you remember it's worth using because we'll use this a lot when we're evaluating sine, cosine and tangent in other quadrants using reference angles.
So remember all three of the trig functions are positive here, only sine is positive here, tangent is positive here, only cosine is positive in the fourth quadrant.
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