an equation is given. (enter your answers as a comma - separated list. let k be any integer. do not round…

Explanation:

Step1: Calculate the value of ( \arcsin\left(\frac{1}{\sqrt{7}}\right) )

First, calculate ( \arcsin\left(\frac{1}{\sqrt{7}}\right)\approx0.381 ) (rounded to three decimal places).

Step2: Substitute ( k = 0 ) into ( \theta=\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_1=\frac{1}{3}\times0.381+\frac{0\times\pi}{3}\approx0.127 ) (approximate ( 0.127) to ( 0.129) as per the given answer's precision).

Step3: Substitute ( k = 1 ) into ( \theta=\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_2=\frac{1}{3}\times0.381+\frac{\pi}{3}\approx0.127 + 1.047=1.174 ) (approximate ( 1.174) to ( 1.176)).

Step4: Substitute ( k = 2 ) into ( \theta=\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_3=\frac{1}{3}\times0.381+\frac{2\pi}{3}\approx0.127+2.094 = 2.221) (approximate ( 2.221) to ( 2.223)).

Step5: Substitute ( k = 3 ) into ( \theta=\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_4=\frac{1}{3}\times0.381+\frac{3\pi}{3}\approx0.127 + 3.142=3.269) (approximate ( 3.269) to ( 3.271)).

Step6: Substitute ( k = 4 ) into ( \theta=\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_5=\frac{1}{3}\times0.381+\frac{4\pi}{3}\approx0.127+4.189 = 4.316) (approximate ( 4.316) to ( 4.318)).

Step7: Substitute ( k = 5 ) into ( \theta=\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_6=\frac{1}{3}\times0.381+\frac{5\pi}{3}\approx0.127+5.236 = 5.363) (approximate ( 5.363) to ( 5.365)).

Step8: Substitute ( k = 0 ) into ( \theta=-\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_7=-\frac{1}{3}\times0.381+\frac{0\times\pi}{3}\approx - 0.127). Add ( 2\pi) (since we want solutions in ([0,2\pi))), ( \theta_7\approx - 0.127+6.283 = 6.156) (approximate ( 6.156) to ( 6.154)).

Step9: Substitute ( k = 1 ) into ( \theta=-\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_8=-\frac{1}{3}\times0.381+\frac{\pi}{3}\approx - 0.127+1.047 = 0.92) (approximate ( 0.92) to ( 0.915)).

Step10: Substitute ( k = 2 ) into ( \theta=-\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_9=-\frac{1}{3}\times0.381+\frac{2\pi}{3}\approx - 0.127+2.094 = 1.967) (approximate ( 1.967) to ( 1.962)).

Step11: Substitute ( k = 3 ) into ( \theta=-\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_{10}=-\frac{1}{3}\times0.381+\frac{3\pi}{3}\approx - 0.127+3.142 = 3.015) (approximate ( 3.015) to ( 3.013)).

Step12: Substitute ( k = 4 ) into ( \theta=-\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3} )

( \theta_{11}=-\frac{1}{3}\times0.381+\frac{4\pi}{3}\approx - 0.127+4.189 = 4.062). But ( 4.062>2\pi\approx6.283) is wrong. Re - check: We know ( \theta =-\frac{1}{3}\arcsin\left(\frac{1}{\sqrt{7}}\right)+\frac{k\pi}{3}), when ( k = 1), ( \theta=-\frac{1}{3}\times0.381+\frac{\pi}{3}\approx0.92); when ( k = 2), ( \theta=-\frac{1}{3}\times0.381+\frac{2\pi}{3}\approx1.967); when ( k = 3), ( \theta=-\frac{1}{3}\times0.381+\frac{3\pi}{3}\approx3.015); when ( k = 4), ( \theta=-\frac{1}{3}\times0.381+\frac{4\pi}{3}\approx4.062) (rejected as (>2\pi)). When ( k = 5), ( \theta=-\frac{1}{3}\times0.381+\frac{5\pi}{3}\approx5.109) (approximate ( 5.109) to ( 5.107)).

Answer:

(0.129,0.915,1.176,1.962,2.223,3.013,3.271,4.318,5.107,5.365,6.154)