Question

if cotθ= 7 and π < θ < 2π , sketch the angle θ and find the value of the other five trig functions

Answer 1

$\cot\theta=7$, so you immediately get

$\dfrac1{\cot\theta}=\boxed{\tan\theta=\dfrac17}$

Recall the Pythagorean identity:

$\tan^2\theta+1=\sec^2\theta\implies\sec\theta=\pm\sqrt{1-\left(\dfrac17\right)^2}=\pm\dfrac{4\sqrt3}7$

and from this we also get cosine for free, since

$\dfrac1{\sec\theta}=\cos\theta=\pm\dfrac7{4\sqrt3}$

But only one of these can be correct. By definition of tangent,

$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$

For $\theta$ between π and 2π, we expect $\sin\theta$ to be negative. We konw $\tan\theta$ is positive, which means $\cos\theta$ must also be negative. So we have

$\boxed{\cos\theta=\dfrac7{4\sqrt3}}$

$\boxed{\sec\theta=\dfrac{4\sqrt3}7}$

and we can find sine using the tangent:

$\tan\theta=\dfrac{\sin\theta}{\cos\theta}\implies\dfrac{\frac7{4\sqrt3}}7=\boxed{\sin\theta=\dfrac1{4\sqrt3}}$

and for free we get

$\dfrac1{\sin\theta}=\boxed{\csc\theta=4\sqrt3}$