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The Tangent Function - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

In right triangle trigonometry (for acute angles only), the tangent is defined as the ratio of the opposite side to the adjacent side. The unit circle definition is tan(theta)=y/x or tan(theta)=sin(theta)/cos(theta). The tangent function is negative whenever sine or cosine, but not both, are negative: the second and fourth quadrants. Tangent is also equal to the slope of the terminal side.

We talked about the sine and cosine functions. Now we want to talk about the tangent function. Tangent function was defined in right triangle trigonometry this way. Tangent theta equals the side opposite theta divided by the side adjacent to theta. So this is theta. Tangent theta is this length divided by this length or y over x. But this definition only works for acute angles, angles between 0 and 90 degrees, because it's only defined this way on right triangles. So we need to find an extension of this definition that works for all angles. Like we have for the sine and cosine.
And that brings us to the unit circle. This is the unit circle definition of tangent. Remember, if I have an angle theta that's drawn in standard position so that its initial side is drawn the positive of x axis and its terminal side crosses the circle at point p. The tangent of theta is defined to be y over x where y and x are these coordinates. So the second coordinate divided by the first coordinate, that's the tangent of theta.
Let's practice with this. This problem asks me to find the tangent of theta for each of these. Well, the tangent of theta here, is going to be the y coordinate divided by the x coordinate. So negative three fifths over negative four fifths. Well that's three quarters. Tangent theta equals three quarters. Here, what's the tangent of theta? We have the 2 coordinates x and y, tangent theta is the y coordinate 12 over 13 divided by the x coordinate -5 over 13. This is going to give me 12 over -5 but -12 over 5. So the tangent theta is -12 over 5. Again this is the unit circle definition of tangent. y over x where y and x are the coordinates of point p.

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