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a.) sec(x + pi/6) = 1 1/cos(x + pi/6) = 1 cos(x + pi/6) = 1
Draw the graph for cosine. Wherever cosine x = 1, so does its reciprocal, secant! -2π, 1, and 2π
So, take those three numbers and subtract π/6
-2π - π/6 is out of the domain, so throw that one away. The other two should be okay. I am answering fast and not checking my arithmetic, but remember that cosine and secant equal each other at the points I gave.
Thanks for the medal, Gonzales
one correction. when you said that the solutions for cos(0) were -2(pi), 1, and 2(pi), i think you meant 0 :)
Yes -- my error! cos (0) = 1. Notice that the problem asks for sec(x+ π/6). That means (I believe) that you need to subtract π/6 from each of those points. Be careful not to go out of the domain, though (-2π to 2π).
@EulersEquation just wanted to know why \[2\pi-(\pi/6) \] is not in the domain will the value no be less than 2 pi
2π - (π/6) = (11π)/6, which is in the domain, and is one of your answers. The other answer is 0 - π/6 = -π/6 However, -2π - (π/6) = (-13π)/6 is just outside the domain to the left.
so is -pi/6 not part of the domain as well @EulersEquation
It is, and is one of your answers.
but how is -pi/6 in the domain @EulersEquation sorry for hassling u again and again really need help
The domain is given as -2π <= x <= 2π. So, -π/6 falls within that interval. does it not?
i am not sure @EulersEquation if u don't mind could u pls explaint o me how -pi/6 fall in the domain i have just started this topic and i am not very confident with it i would be really grateful if u could explain this to me.
The domain is given by the interval -2π <= x <= 2π Since -π/6 is greater than -2π and is less than 2π, it falls within the given interval.
oh i get it thank you @EulersEquation
@EulersEquation so for the second part of the question will the answers be -pi/2 and -3pi/2
The inverse of csc is sin, so you can use the sine graph. sin (-π/2) = -1, so in order for sin(x/3) = -1, x/3 = -π/2. Solving for x gives x = -3π/2. The other place in the interval where sin x = -1 is at sin(3π/2). Solving the equation x/3 = 3π/2, gives x = 9π/2, which is clearly outside the domain. So the only answer is x = -3π/2. I think (you may need to check my arithmetic).