Question

Given coso= 4/9 and csc 0<0, find sin0 and tan0

Answer 1

`\sin\theta=-(√(65))/(9).`

`\tan\theta=-(√(65))/(4).`

Explanation:

1. Use the man trigonometric equality

`\cos^2\theta+\sin^2\theta=1.`

From this equality

`\sin^2\theta=1-\cos^2\theta,\\ \\\sin^2\theta=1-\left((4)/(9)\right)^2,\\ \\\sin^2\theta=1-(16)/(81),\\ \\\sin^2\theta=(65)/(81).`

2. Since
`\csc\theta=(1)/(\sin\theta)<0,` you can state that
`\sin\theta<0` and

`\sin\theta=-\sqrt{(65)/(81)}=-(√(65))/(9).`

3. Use the definition:

`\tan\theta=(\sin\theta)/(\cos\theta)=-((√(65))/(9))/((4)/(9))=-(√(65))/(4).`