Question

Find the exact value by using a half angle identity.tan 75°

Answer 1

The trigonometric function is given as

`\tan 75^(\circ)`

Apply the half angle identity to find the value of tan 75 ,

`\tan ((u)/(2))=(\sin u)/(1+\cos u)`

Here,

`\tan ((150^(\circ))/(2))=(\sin150^(\circ))/(1+cos150^(\circ))`
`\tan (75^(\circ))=\frac{(1)/(2)}{1-\frac{\sqrt[]{3}}{2}}=\frac{(1)/(2)}{\frac{2-\sqrt[]{3}}{2}}^{}`
`\tan 75^(\circ)=\frac{1}{2-\sqrt[]{3}}`

Now rationalize the function.

`\tan 75^(\circ)=\frac{1}{2-\sqrt[]{3}}*\frac{2+\sqrt[]{3}}{2+\sqrt[]{3}}=\frac{2+\sqrt[]{3}}{4-3}=\frac{2+\sqrt[]{3}}{1}`

Again simplify the trigonometric function,

`\tan 75^(\circ)=2+1.732=3.732`

Hence the answer is 3.732.