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Solving Trigonometric Equations - Problem 2

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.

I want to show you an example of a tangent equation. I have tangent of x over 4 or equals -1. This equation is also different because this is our first example where we’ve got something other than just x of the variable inside the tangent function and I actually want to show you how to deal with that.

I like to make little substitution and turn in into a simple trigonometric function, so I’m going to call this x over 4 theta and that will give me tangent of theta equals -1. So first I’m going to look for angles theta that satisfy this equation and then I’ll re-substitute x over 4 for theta in the very end.

When I’m looking for angles whose tangents is -1, I turn to the unit circle and I draw myself an angle. Actually you can draw yourself a line because remember that the tangent of an angle represents, for example the tangent of this angle represents the slope of segment OP. So what we are looking for is an angle that gives us a terminal side with slope -1.

Now the point that we are looking for pi root 2 over 2 combine negative root 2 over 2 and this corresponds to negative pi over 4. This is precisely the answer that you would get if you use inverse tangent. Theta equals inverse tangent of -1 that’ll give you negative pi over 4. So that answer being the principle solution of our equation, now unlike sine and cosine there’s actually going to be one solution per period of tangents so this negative pi that’s our only principle solution. And so to get the rest of them we use periodicity of tangent. And that’s where tangent is different again.

Tangent has period pi so to get all the solutions I need to say theta equals minus pi over 4 plus n pi. So you could add any integer of multiple of pi to negative pi over 4 and get another solution if I add a single pi I get negative pi over 4 plus pi 3 pi over 4 and if you look at the unit circle that’s this angle. And so on you could add another pi, 2 more, as many as you like or subtract.

So this represents all the solutions to the equation tangent of theta equals -1 but remember we made a substitution, theta equals x over 4. So x over 4 equals negative pi over 4 plus nPi to get the final solutions for x we multiply everything by 4 and we get x equals negative pi plus 4 n pi and that’s our final solution.

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