Trigonometric Equations

Summary

The solutions of tan x = k are	x =	... , tan-1 k , π + tan-1 k , 2π + tan-1 k ,
		3π + tan-1 k , 4π + tan-1 k , 5π + tan-1 k , ...
Therefore the general solution of tan x = k is	x =	nπ + tan-1 k
	=	180°n + tan-1 k      for all integer n

The solutions of cos x = k are	x =	... , - cos-1 k , cos-1 k , 2π - cos-1 k ,
		2π + cos-1 k , 4π - cos-1 k , 4π + cos-1 k ,
		6π - cos-1 k , 6π + cos-1 k , ...
Therefore the general solution of cos x = k is	x =	2nπ ± cos-1 k
	=	360°n ± cos-1 k      for all integer n

The solutions of sin x = k are	x =	... , sin-1 k , π - sin-1 k , 2π + sin-1 k ,
		3π - sin-1 k , 4π + sin-1 k , 5π - sin-1 k , ...
Therefore the general solution of sin x = k is	x =	nπ + (-1)n sin-1 k
	=	180°n + (-1)n sin-1 k      for all integer n

Exercises for students
Find the solutions of the following equations (in degrees & in radians) in the range 0^o < x < 360^o and check your answers using the above applet.
Write down the general solutions:

tan x = 1

cos x = 0.5

sin x = 0.5

tan x = -1

cos x = -0.5

sin x = -0.5

Find the general solution, in degrees, of the equation tan 3x = tan 30°.

Find the general solution, in degrees, of the equation cos 2x = 0.6 .

Find the general solution, in radians, of the equation sin 3x = 1/2.

Find all the solution of tan 2θ = 1 in 0 < θ < 360°.

Find the general solution of the equation sin 2x = cos 30°.


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