how do you determine the solutions?

Question

Determine the solution(s) to the equation root3tantheta + 3 = 1 where 0</=theta</=2pi.

Answer 1

For this problem, we'll use algebra to simplify the problem to tan theta = something, and then take the inverse tan to solve for the angle.

 

(root 3) tan theta + 3 = 1

(root 3) tan theta = -2

tan theta = -2/(root 3)

tan^-1 (tan theta) = tan^-1 (-2/(root 3))

theta = -49.1º

 

In which quadrants is the tan function negative?  A special way to remember this is to remember ASTC, or All Sine Tangent Cosine. (Visual representation: http://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/All_Students_Take_Calculus.svg/220px-All_Students_Take_Calculus.svg.png)

 

In the first quadrant, all the functions (sine, tangent, cosine) are all positive

In the second quadrant, only sine is positive

In the third quadrant, only tangent is positive

In the fourth quadrant, only cosine is positive

 

The answer we got is negative for tangent.  So going back to the question, in which quadrants is the tan function negative, we get the 2nd quadrant and the fourth quadrant.

 

To get the angle for the 2nd quadrant, we do 180 + (our angle) = 180 + (-49.1) = 130.1

To get the angle of the 4th quadrant, we do 360 + (our angle) = 360 + (-49.1) = 310.1

 

Therefore, the two answers we have for root3tantheta + 3 = 1 where 0</=theta</=2pi are 130.1º and 310.1º.

 

If you have any questions, feel free to ask!

Answer 2

Root3 tantheta+3=1

 

Root3 tantheta=-2

 

tantheta= -2/root3

 

tanthets= -1.1547

 

theta = 130.893 degrees or  2.285 rad

 

   OR. = 310.893 degrees or 5.428 rad

 

Since tantheta is negative in second and fourth quadrant.