Question

If sin theta= - 5/7, which of the following are possible?

Answer 1

`\large\boxed{\sec\theta=(7)/(√(24)),\ and\ \tan\theta=-(5)/(√(24))}\\\boxed{\cos\theta=(√(24))/(7),\ and\ \tan\theta=(5)/(√(24))}`

Explanation:

`\text{Use:}\\\\\sin^2\theta+\cos^2\theta=1\to\cos\theta=\pm√(1-\sin^2\theta)\\\\\tan\theta=(\sin\theta)/(\cos\theta)\\\\\sec\theta=(1)/(\cos\theta)\\\\==========================\\\\\sin\theta=-(5)/(7)\\\\\cos\theta=\pm\sqrt{1-\left(-(5)/(7)\right)^2}=\pm\sqrt{1-(25)/(49)}=\pm\sqrt{(24)/(49)}=\pm(√(24))/(√(49))=\pm(√(24))/(7)\\\\\sec\theta=\pm(1)/((√(24))/(7))=\pm(7)/(√(24))`

`\tan\theta=\pm(-(5)/(7))/((√(24))/(7))=\mp(5)/(7\!\!\!\!\diagup_1)\cdot(7\!\!\!\!\diagup^1)/(√(24))=\mp(5)/(√(24))`

`\sin\theta=-(5)/(7)<0\to\text{III or IV Quadrant}\\\\\text{Look at the picture}\\\\\text{Quadrant III}:\\\\\sin<0,\ \cos<0,\ \sec<0,\ \tan>0\\\\\text{Quadrant IV}:\\\\\sin<0,\ \cos>0,\ \sec>0,\ \tan<0`