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Evaluating the Tangent Function - Concept 10,397 views

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.

When evaluating the tangent function, to find values of the tangent function at different angles, we first identify the reference angle formed by the terminal side and the x-axis. Then, we find the tangent of this reference angle and, based on which quadrant the terminal side is in, decide if tangent is positive or negative. Tangent is positive in the first and third quadrants, where both sine and cosine are positive and both are negative.

Okay, we just learned some basic values of the tangent function in the first quadrant. And I want to show you how to find values of tangent in other quadrants. For example find the tangent of 3 pi over 4.
The first step is to draw a diagram. Done. The second step is to identify the reference angle. Now the reference angle is the angle between the terminal side and the x axis. In this case, pi over 4. Once you've identified the reference angle then you want to take, oops. Sorry. You want to take the tangent, the tangent of that angle. That's our first quadrant angle and its tangent is 1. Once you've got that, the tangent of 3 pi over 4 is either plus or minus this value. It's plus or minus depending on which quadrant you're in. And you have to remember the mnemonic all students take calculus. We're in the second quadrant right now, where only sine is positive. the others are negative. So tangent's negative here and that means tangent of root 3 over 4 is -1.
So just remember draw a diagram, identify the reference angle, find the tangent of that reference angle and then use the quadrant to decide whether the tangent's positive or negative.

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