Question

Find cos x and cot x if sin x = 1/4 and tan x < 0.

Answer 1

For this we use the trigonometric identities

`\begin{gathered} \cos ^2(x)+\sin ^2(x)=1 \\ \tan (x)=(\sin (x))/(\cos (x)) \\ \cot (x)=(\cos (x))/(\sin (x)) \end{gathered}`

Since we have the value of sin (x) we start from the first identity to obtain cos (x):

`\begin{gathered} \sin (x)=(1)/(4)\Rightarrow\sin ^2(x)=(1)/(4^2)\Rightarrow\sin ^2(x)=(1)/(16) \\ \text{Then,} \\ \cos ^2(x)+\sin ^2(x)=1 \\ \cos ^2(x)+(1)/(16)=1 \\ \cos ^2(x)=1-(1)/(16) \\ \cos ^2(x)=(15)/(16) \\ \cos (x)=\pm\sqrt[]{(15)/(16)} \\ \text{ Since tangent is negative and by the second propery, we take} \\ \cos (x)=-\sqrt[]{(15)/(16)} \\ \cos (x)=\frac{-\sqrt[]{15}}{\sqrt[]{16}} \\ \cos (x)=\frac{-\sqrt[]{15}}{4} \end{gathered}`

Finally, we use the third property to obtain cot (x):

`\begin{gathered} \cot (x)=(\cos(x))/(\sin(x)) \\ \cot (x)=\frac{\frac{-\sqrt[]{15}}{4}}{(1)/(4)} \end{gathered}`

Applying the Sandwich Law for Fractions, we have

`\begin{gathered} \cot (x)=\frac{-\sqrt[]{15}\cdot4}{4\cdot1} \\ \cot (x)=-\sqrt[]{15} \end{gathered}`