Question

If sin θ = 3 over 7 and tan θ < 0, what is the value of cos θ?

Answer 1

`if\ \sin\Theta \ \textgreater \ 0\ and\ \tan\Theta \ \textless \ 0\ then\ \Theta\in\underbrace{(90^o;\ 180^o)}_{\left((\pi)/(2);\ \pi\right)}\\\\therefore\ \cos\Theta \ \textless \ 0.`


`\text{Use:}\ \sin^2x+\cos^2x=1\\\\\left((3)/(7)\right)^2+\cos^2\Theta=1\\\\(9)/(49)+\cos^2\Theta=1\ \ \ |-(9)/(49)\\\\\cos^2\Theta=(40)/(49)\to\cos\Theta=-\sqrt{(40)/(49)}\\\\\cos\Theta=-(√(40))/(√(49))\\\\\cos\Theta=-(√(4\cdot10))/(7)\\\\\boxed{\cos\Theta=-(2√(10))/(7)}`