Question

If sinA+cosecA=3 find the value of sin2A+cosec2A​

Answer 1

Answer:

$\sin 2A + \csc 2A = 2.122$

Step-by-step explanation:

Let $f(A) = \sin A + \csc A$, we proceed to transform the expression into an equivalent form of sines and cosines by means of the following trigonometrical identity:

$\csc A = \frac{1}{\sin A}$ (1)

$\sin^{2}A +\cos^{2}A = 1$ (2)

Now we perform the operations: $f(A) = 3$

$\sin A + \csc A = 3$

$\sin A + \frac{1}{\sin A} = 3$

$\sin ^{2}A + 1 = 3\cdot \sin A$

$\sin^{2}A -3\cdot \sin A +1 = 0$ (3)

By the quadratic formula, we find the following solutions:

$\sin A_{1} \approx 2.618$ and $\sin A_{2} \approx 0.382$

Since sine is a bounded function between -1 and 1, the only solution that is mathematically reasonable is:

$\sin A \approx 0.382$

By means of inverse trigonometrical function, we get the value associate of the function in sexagesimal degrees:

$A \approx 22.457^{\circ}$

Then, the values of the cosine associated with that angle is:

$\cos A \approx 0.924$

Now, we have that $f(A) = \sin 2A +\csc2A$, we proceed to transform the expression into an equivalent form with sines and cosines. The following trignometrical identities are used:

$\sin 2A = 2\cdot \sin A\cdot \cos A$ (4)

$\csc 2A = \frac{1}{\sin 2A}$ (5)

$f(A) = \sin 2A + \csc 2A$

$f(A) = \sin 2A + \frac{1}{\sin 2A}$

$f(A) = \frac{\sin^{2} 2A+1}{\sin 2A}$

$f(A) = \frac{4\cdot \sin^{2}A\cdot \cos^{2}A+1}{2\cdot \sin A \cdot \cos A}$

If we know that $\sin A \approx 0.382$ and $\cos A \approx 0.924$, then the value of the function is:

$f(A) = \frac{4\cdot (0.382)^{2}\cdot (0.924)^{2}+1}{2\cdot (0.382)\cdot (0.924)}$

$f(A) = 2.122$