Question

How do I find these exact values with the given information and values?

Answer 1

`\begin{gathered} \sin 2x=(12)/(13) \\ \cos 2x=-(5)/(13) \\ \tan 2x=-(12)/(5) \end{gathered}`

Explanation:

We know that cotagent is given as follows:

`\begin{gathered} \cot x=\frac{\text{ adjacent }}{\text{ opposite}} \\ \text{ therefore} \\ \text{adjacent = 2} \\ \text{ oppoiste = 3} \\ \text{ hypotenuse =}\sqrt[]{2^2+3^2}=\sqrt[]{4+9}=\sqrt[]{13} \end{gathered}`

Therefore:

sin 2x:

`\begin{gathered} \sin 2x=2\sin x\cdot\cos x \\ \sin x=\frac{\text{ opposite}}{\text{ hypotenuse}}=\frac{3}{\sqrt[]{13}} \\ \cos x=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{2}{\sqrt[]{13}} \\ \sin 2x=2\sin x\cdot\cos x=2\cdot(3)/(√(13))\cdot(2)/(√(13))=(12)/(13) \end{gathered}`

cos 2x:

`\begin{gathered} \cos 2x=\cos ^2x-\sin ^2x \\ \sin ^2x=\mleft((3)/(√(13))\mright)^2=(9)/(13) \\ \cos ^2x=\mleft((2)/(√(13))\mright)^2=(4)/(13) \\ \cos 2x=(4)/(13)-(9)/(13)=-(5)/(13) \end{gathered}`

tan 2x:

`\tan 2x=(\sin 2x)/(\cos 2x)=((12)/(13))/(-(5)/(13))=-(12)/(5)`