Question

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

Answer 1

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

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 = (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 + (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 = (1)/(\sin 2A)` (5)

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

`f(A) = \sin 2A + (1)/(\sin 2A)`

`f(A) = (\sin^(2) 2A+1)/(\sin 2A)`

`f(A) = (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) = (4\cdot (0.382)^(2)\cdot (0.924)^(2)+1)/(2\cdot (0.382)\cdot (0.924))`

`f(A) = 2.122`