Question

Sin∅=√3-1/2 find approximate value of sec∅(sec∅+tan∅)/1+tan²∅​

Answer 1

The approximate value of
`f(\theta) = (\sec \theta \cdot (\sec \theta+\tan \theta))/(1+\tan^(2)\theta)` is 1.366.

Explanation:

Let
`f(\theta) = (\sec \theta \cdot (\sec \theta+\tan \theta))/(1+\tan^(2)\theta)`, we proceed to simplify the formula until a form based exclusively in sines and cosines is found. From Trigonometry, we shall use the following identities:

`\sec \theta = (1)/(\cos \theta)` (1)

`\tan\theta = (\sin\theta)/(\cos \theta)` (2)

`\cos^(2)+\sin^(2) = 1` (3)

Then, we simplify the given formula:

`f(\theta) = (\left((1)/(\cos \theta) \right)\cdot \left((1)/(\cos \theta)+(\sin \theta)/(\cos \theta)\right) )/(1+(\sin^(2)\theta)/(\cos^(2)\theta) )`

`f(\theta) = \frac{\left((1)/(\cos^(2) \theta) \right)\cdot (1+\sin \theta)}{\frac{\sin^(2)\theta + \cos^2{\theta}}{\cos^(2)\theta} }`

`f(\theta) = (\left((1)/(\cos^(2)\theta)\right)\cdot (1+\sin \theta))/((1)/(\cos^(2)\theta) )`

`f(\theta) = 1+\sin \theta`

If we know that
`\sin \theta =(√(3)-1)/(2)`, then the approximate value of the given function is:

`f(\theta) = 1 +(√(3)-1)/(2)`

`f(\theta) = (√(3)+1)/(2)`

`f(\theta) \approx 1.366`