Answer:
[tex]\sin 2A + \csc 2A = 2.122[/tex]
Step-by-step explanation:
Let [tex]f(A) = \sin A + \csc A[/tex], we proceed to transform the expression into an equivalent form of sines and cosines by means of the following trigonometrical identity:
[tex]\csc A = \frac{1}{\sin A}[/tex] (1)
[tex]\sin^{2}A +\cos^{2}A = 1[/tex] (2)
Now we perform the operations: [tex]f(A) = 3[/tex]
[tex]\sin A + \csc A = 3[/tex]
[tex]\sin A + \frac{1}{\sin A} = 3[/tex]
[tex]\sin ^{2}A + 1 = 3\cdot \sin A[/tex]
[tex]\sin^{2}A -3\cdot \sin A +1 = 0[/tex] (3)
By the quadratic formula, we find the following solutions:
[tex]\sin A_{1} \approx 2.618[/tex] and [tex]\sin A_{2} \approx 0.382[/tex]
Since sine is a bounded function between -1 and 1, the only solution that is mathematically reasonable is:
[tex]\sin A \approx 0.382[/tex]
By means of inverse trigonometrical function, we get the value associate of the function in sexagesimal degrees:
[tex]A \approx 22.457^{\circ}[/tex]
Then, the values of the cosine associated with that angle is:
[tex]\cos A \approx 0.924[/tex]
Now, we have that [tex]f(A) = \sin 2A +\csc2A[/tex], we proceed to transform the expression into an equivalent form with sines and cosines. The following trignometrical identities are used:
[tex]\sin 2A = 2\cdot \sin A\cdot \cos A[/tex] (4)
[tex]\csc 2A = \frac{1}{\sin 2A}[/tex] (5)
[tex]f(A) = \sin 2A + \csc 2A[/tex]
[tex]f(A) = \sin 2A + \frac{1}{\sin 2A}[/tex]
[tex]f(A) = \frac{\sin^{2} 2A+1}{\sin 2A}[/tex]
[tex]f(A) = \frac{4\cdot \sin^{2}A\cdot \cos^{2}A+1}{2\cdot \sin A \cdot \cos A}[/tex]
If we know that [tex]\sin A \approx 0.382[/tex] and [tex]\cos A \approx 0.924[/tex], then the value of the function is:
[tex]f(A) = \frac{4\cdot (0.382)^{2}\cdot (0.924)^{2}+1}{2\cdot (0.382)\cdot (0.924)}[/tex]
[tex]f(A) = 2.122[/tex]
