Question

If x = 4 tan(θ), find sec(θ) in terms of x

Answer 1

`\sec \theta = \pm \frac{ \sqrt{ {x}^(2) + 16} }{4}`

Explanation:

`\because \: x = 4 \tan \theta \\ \therefore(x)/(4) = \tan \theta....(1) \\ \\ \because \: { \sec}^(2) \theta = 1 + { \tan}^(2) \theta \\ \therefore \: \sec \theta = \pm\sqrt{1 + { \tan}^(2) \theta } \\ \therefore \: \sec \theta = \pm\sqrt{1 + { \bigg( (x)/(4) \bigg)}^(2) } \\ \therefore \: \sec \theta = \pm\sqrt{1 + { (x^(2))/(16) } } \\ \therefore \: \sec \theta = \pm\sqrt{{ (16 + x^(2))/(16) } } \\ \therefore \: \sec \theta = \pm \frac{ \sqrt{ {x}^(2) + 16} }{4}`