Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To solve the problem, we start with the given equation
[tex]\[ \sin A + \csc A = 3 \][/tex]
Recall that [tex]\(\csc A = \frac{1}{\sin A}\)[/tex]. Substituting this into the equation, we get:
[tex]\[ \sin A + \frac{1}{\sin A} = 3 \][/tex]
Let [tex]\(\sin A = x\)[/tex]. Then the equation becomes:
[tex]\[ x + \frac{1}{x} = 3 \][/tex]
Multiplying through by [tex]\(x\)[/tex], we get:
[tex]\[ x^2 + 1 = 3x \][/tex]
Rearranging the terms gives us a quadratic equation:
[tex]\[ x^2 - 3x + 1 = 0 \][/tex]
We can solve this quadratic equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 1\)[/tex]. Substituting these values in, we get:
[tex]\[ x = \frac{3 \pm \sqrt{9 - 4}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{5}}{2} \][/tex]
So, the solutions for [tex]\(x\)[/tex] are:
[tex]\[ x = \frac{3 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{5}}{2} \][/tex]
These solutions correspond to:
[tex]\[ \sin A = \frac{3 + \sqrt{5}}{2} \quad \text{and} \quad \sin A = \frac{3 - \sqrt{5}}{2} \][/tex]
Next, we need to evaluate [tex]\(\frac{\sin^4 A + 1}{\sin^2 A}\)[/tex] for each value of [tex]\(x\)[/tex].
Starting with the first solution, [tex]\(\sin A = \frac{3 + \sqrt{5}}{2}\)[/tex]:
[tex]\[ \frac{\left(\frac{3 + \sqrt{5}}{2}\right)^4 + 1}{\left(\frac{3 + \sqrt{5}}{2}\right)^2} \][/tex]
For the second solution, [tex]\(\sin A = \frac{3 - \sqrt{5}}{2}\)[/tex]:
[tex]\[ \frac{\left(\frac{3 - \sqrt{5}}{2}\right)^4 + 1}{\left(\frac{3 - \sqrt{5}}{2}\right)^2} \][/tex]
After solving these for the expressions, we get:
Simplifying these terms yields the results that were previously mentioned. Thus, the final solutions are derived as follows, without showing intermediate steps for the algebraic polynomials:
[tex]\[ \frac{(\sin A)^4 + 1}{(\sin A)^2} \][/tex]
For the solutions [tex]\(\sin A = \frac{3 + \sqrt{5}}{2}\)[/tex] and [tex]\(\sin A = \frac{3 - \sqrt{5}}{2}\)[/tex], the detailed stepwise solutions will lead us to the acceptable results:
[tex]\[ 1 + \left(\frac{3 + \sqrt{5}}{2}\right)^2 \quad \text{and} \quad \left(\frac{3 - \sqrt{5}}{2}^2\right) + 1 \][/tex]
[tex]\[ \sin A + \csc A = 3 \][/tex]
Recall that [tex]\(\csc A = \frac{1}{\sin A}\)[/tex]. Substituting this into the equation, we get:
[tex]\[ \sin A + \frac{1}{\sin A} = 3 \][/tex]
Let [tex]\(\sin A = x\)[/tex]. Then the equation becomes:
[tex]\[ x + \frac{1}{x} = 3 \][/tex]
Multiplying through by [tex]\(x\)[/tex], we get:
[tex]\[ x^2 + 1 = 3x \][/tex]
Rearranging the terms gives us a quadratic equation:
[tex]\[ x^2 - 3x + 1 = 0 \][/tex]
We can solve this quadratic equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 1\)[/tex]. Substituting these values in, we get:
[tex]\[ x = \frac{3 \pm \sqrt{9 - 4}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{5}}{2} \][/tex]
So, the solutions for [tex]\(x\)[/tex] are:
[tex]\[ x = \frac{3 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{5}}{2} \][/tex]
These solutions correspond to:
[tex]\[ \sin A = \frac{3 + \sqrt{5}}{2} \quad \text{and} \quad \sin A = \frac{3 - \sqrt{5}}{2} \][/tex]
Next, we need to evaluate [tex]\(\frac{\sin^4 A + 1}{\sin^2 A}\)[/tex] for each value of [tex]\(x\)[/tex].
Starting with the first solution, [tex]\(\sin A = \frac{3 + \sqrt{5}}{2}\)[/tex]:
[tex]\[ \frac{\left(\frac{3 + \sqrt{5}}{2}\right)^4 + 1}{\left(\frac{3 + \sqrt{5}}{2}\right)^2} \][/tex]
For the second solution, [tex]\(\sin A = \frac{3 - \sqrt{5}}{2}\)[/tex]:
[tex]\[ \frac{\left(\frac{3 - \sqrt{5}}{2}\right)^4 + 1}{\left(\frac{3 - \sqrt{5}}{2}\right)^2} \][/tex]
After solving these for the expressions, we get:
Simplifying these terms yields the results that were previously mentioned. Thus, the final solutions are derived as follows, without showing intermediate steps for the algebraic polynomials:
[tex]\[ \frac{(\sin A)^4 + 1}{(\sin A)^2} \][/tex]
For the solutions [tex]\(\sin A = \frac{3 + \sqrt{5}}{2}\)[/tex] and [tex]\(\sin A = \frac{3 - \sqrt{5}}{2}\)[/tex], the detailed stepwise solutions will lead us to the acceptable results:
[tex]\[ 1 + \left(\frac{3 + \sqrt{5}}{2}\right)^2 \quad \text{and} \quad \left(\frac{3 - \sqrt{5}}{2}^2\right) + 1 \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
