Question

Please help me thank you

Answer 1

`\large\boxed{-(8)/(√(63))}`

Explanation:

`\csc\theta=(1)/(\sin\theta)\\\\\cos\theta=-(1)/(8)\qquad\text{use}\ \sin^2\theta+\cos^2\theta=1\\\\\sin^2\theta+\left(-(1)/(8)\right)^2=1\\\\\sin^2\theta+(1)/(64)=1\qquad\text{subtract}\ (1)/(64)\ \text{from both sides}\\\\\sin^2\theta=(64)/(64)-(1)/(64)\\\\\sin^2\theta=(63)/(64)\to\sin\theta=\pm\sqrt{(63)/(64)}\\\\180^o<\theta<270^o\to III\ quadrant,\ \sin\theta<0.\\\\\text{Therefore}\ \sin\theta=-\sqrt{(63)/(64)}=-(√(63))/(√(64))=-(√(63))/(8).`

`\csc\theta=(1)/(-(√(63))/(8))=-(8)/(√(63))`