Question 1174893: Solve the following trigonometric equations without using a calculator. Find all solutions where 0 ≤ x < 2 π . If no solution exists write "DNE". (Hint: isolate the trig function then draw triangles.)
tan^2(x)-1=0
*(I got π/4,3π/4,and 5π/4, I am not sure if I have all the solutions)*
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52803)   (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
tan^2(x) - 1 = 0
tan^2(x) = 1
tan(x) = sqrt(1) or tan(x) = -sqrt(1)
tan(x) = 1 or tan(x) = -1
If tan(x) = 1, then x = pi/4 or x = 5pi/4
Use the unit circle.
Note the jump from pi/4 to 5pi/4 is exactly pi units
pi/4+pi = 5pi/4
If we know one solution for tangent, then we add or subtract pi units to get to another solution. This is because the period of the tangent function is exactly pi units. It repeats itself every pi units.
If tan(x) = -1, then x = 3pi/4 or x = 7pi/4
3pi/4 + pi = 7pi/4
There are four solutions in the interval  that satisfy tan^2(x) - 1 = 0 and those four solutions are:
pi/4, 3pi/4, 5pi/4, 7pi/4
The nice pattern here is that the coefficients for the pi terms in the numerator is the set {1,3,5,7}
Something like 9pi/4 doesn't work because it's too large
9pi/4 > 2pi
9pi > 4*2pi
9pi > 8pi
9 > 8
In other words, x = 9pi/4 is outside the interval
Or you could note that 9pi/4 = 7.07 approximately and 2pi = 6.28 approximately, to help see that 9pi/4 is too big.
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