Given:
[tex]tan\theta=\frac{1}{7}\text{ and sin}\theta<0[/tex]Required:
Find the value of cos in Radical form.
Explanation:
Use the trigonometric ratio:
[tex]\begin{gathered} tan\theta=\frac{opp.}{adj.} \\ tan\theta=\frac{1}{7} \end{gathered}[/tex]Therefore opp. = 1 and adj. = 7
Find the hypotenuse by using the Pythagoras theorem.
[tex]\begin{gathered} hyp.=\sqrt{(opp.)^2+(adj.)^2} \\ hyp.=\sqrt{(1)^2+(7)^2} \\ hyp.=\sqrt{1+49} \\ hyp.=\sqrt{50} \end{gathered}[/tex]Use the trigonometric ratio for cosine as:
[tex]\begin{gathered} cos\theta=\frac{adj.}{hyp.} \\ cos\theta=\frac{7}{\sqrt{50}} \end{gathered}[/tex]Given that sin < 0 and tan = 1/7 > 0
It will be possible only when cos < 0
So
[tex]cos\theta=-\frac{7}{\sqrt{50}}[/tex]Final Answer:
[tex]cos\theta=-\frac{7}{\sqrt{50}}[/tex]