Question

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

Answer 1

Answer:

$\sec \theta + \csc \theta = \frac{2\cdot \sqrt{3}+6}{3}$

Step-by-step 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) $\sin \theta = \frac{1}{2}$  Compatibility with multiplication/Existence of multiplicative inverse/Modulative property/Definition of division

6) $\theta = \sin^{-1} \frac{1}{2}$ Inverse trigonometric inverse.

7) $\theta = 30^{\circ}$ Result.

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

$\sec \theta = \frac{1}{\cos \theta}$, $\csc \theta = \frac{1}{\sin \theta}$ (1, 2)

In addition, we know that $\sin 30^{\circ} = \frac{1}{2}$ and $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$, then the sum of the two trigonometric function abovementioned is:

$\sec \theta + \csc \theta = \frac{1}{\cos \theta} + \frac{1}{\sin \theta}$

$\frac{2}{\sqrt{3}} + 2 = \frac{2\cdot \sqrt{3}}{3} + 2$

$\sec \theta + \csc \theta = \frac{2\cdot \sqrt{3}+6}{3}$