Question

If cos theta = 4/ square root 23 and angle theta is in quadrant 1, what is the exact value of tan2 theta in simplest radical form?

Answer 1

For this case we know that an angle theta is in the I quadrant and additionally:

`\cos \theta=\frac{4}{\sqrt[]{23}}`

We can find the sin of the angle usingt this identity:

`\sin ^2\theta=1-\cos ^2\theta`
`\sin \theta=\sqrt[]{1-(\frac{4}{\sqrt[]{23}})^2}=\sqrt[]{(7)/(23)}`

We can find tangent:

`\tan \theta=(\sin\theta)/(\cos\theta)=\frac{\sqrt[]{(7)/(23)}}{\frac{4}{\sqrt[]{23}}}=\frac{\sqrt[]{7}}{4}`

Finally we can find tan 2 theta and we have:

`\tan 2\theta=(2\tan\theta)/(1-\tan^2\theta)=\frac{2\cdot\frac{\sqrt[]{7}}{4}}{1-(7)/(16)}=\frac{\frac{\sqrt[]{7}}{2}}{(9)/(16)}=\frac{8\sqrt[]{7}}{9}`