Question

If 7sin²∅+3cos²∅is equal to 4,then evaluate sec∅+cosec∅​

Answer 1

`\sec \theta + \csc \theta = (2\cdot √(3)+6)/(3)`

Explanation:

We proceed to simplify the given trigonometric expression into a form with a single trigonometric function:

1)
`7\cdot \sin^(2)\theta + 3\cdot \cos^(2)\theta = 4` Given.

2)
`4\cdot \sin^(2)\theta + 3\cdot (\sin^(2)\theta + \cos^(2)\theta) = 4` Definition of addition/Associative and distributive properties.

3)
`4\cdot \sin^(2) \theta +3 = 4`
`\sin^(2)\theta + \cos^(2)\theta = 1`/Modulative property

4)
`4\cdot \sin^(2)\theta = 1` Compatibility with addition/Existence of additive inverse/Modulative property

5)
`\sin \theta = (1)/(2)` Compatibility with multiplication/Existence of multiplicative inverse/Modulative property/Definition of division

6)
`\theta = \sin^(-1) (1)/(2)` Inverse trigonometric inverse.

7)
`\theta = 30^(\circ)` Result.

By Trigonometry, we know that secant and cosecant functions have the following identities:

`\sec \theta = (1)/(\cos \theta)`,
`\csc \theta = (1)/(\sin \theta)` (1, 2)

In addition, we know that
`\sin 30^(\circ) = (1)/(2)` and
`\cos 30^(\circ) = (√(3))/(2)`, then the sum of the two trigonometric function abovementioned is:

`\sec \theta + \csc \theta = (1)/(\cos \theta) + (1)/(\sin \theta)`

`(2)/(√(3)) + 2 = (2\cdot √(3))/(3) + 2`

`\sec \theta + \csc \theta = (2\cdot √(3)+6)/(3)`