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BaLLatris [955]
3 years ago
12

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

Mathematics
1 answer:
valina [46]3 years ago
8 0

\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}

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