Question
The equation 3 sine x plus cosine squared x equals 2 is solved below.
math expression3 sin x plus cosine squared x equals 2 line break
3 sin x plus left parenthesis 2 minus sine squared x right parenthesis equals 2 line break
3 sin x minus sine squared x plus 1 equals 2 line break
3 sin x minus sine squared x minus 1 equals 0 line break
sine squared x minus 3 sin x plus 1 equals 0 line break
left parenthesis sine x minus 3 right parenthesis left parenthesis sine x minus 1 right parenthesis equals 0 line break
sine x equals 3 or sine x equals negative 1
x equals sine superscript negative 1 baseline left parenthesis 3 right parenthesis or x equals sine superscript negative 1 baseline left parenthesis negative 1 right parenthesis line break
x equals start fraction 2 pi over 3 end fraction plus 2 lower n pi
Which of the following best describes the solution?
(1 point)
Responses
The Pythagorean identity was applied incorrectly.
The Pythagorean identity was applied incorrectly.
The equation was factored incorrectly.
The equation was factored incorrectly.
The inverses were taken incorrectly.
The inverses were taken incorrectly.
There is no error in the solution.
math expression3 sin x plus cosine squared x equals 2 line break
3 sin x plus left parenthesis 2 minus sine squared x right parenthesis equals 2 line break
3 sin x minus sine squared x plus 1 equals 2 line break
3 sin x minus sine squared x minus 1 equals 0 line break
sine squared x minus 3 sin x plus 1 equals 0 line break
left parenthesis sine x minus 3 right parenthesis left parenthesis sine x minus 1 right parenthesis equals 0 line break
sine x equals 3 or sine x equals negative 1
x equals sine superscript negative 1 baseline left parenthesis 3 right parenthesis or x equals sine superscript negative 1 baseline left parenthesis negative 1 right parenthesis line break
x equals start fraction 2 pi over 3 end fraction plus 2 lower n pi
Which of the following best describes the solution?
(1 point)
Responses
The Pythagorean identity was applied incorrectly.
The Pythagorean identity was applied incorrectly.
The equation was factored incorrectly.
The equation was factored incorrectly.
The inverses were taken incorrectly.
The inverses were taken incorrectly.
There is no error in the solution.
Answers
GPT-4o mini
To analyze the solution provided, let's break it down step-by-step.
1. The original equation is \(3 \sin x + \cos^2 x = 2\).
2. The transformation using the Pythagorean identity \(\cos^2 x = 1 - \sin^2 x\) is correctly applied to yield \(3 \sin x + (2 - \sin^2 x) = 2\).
3. Rearranging leads to the equation \(3 \sin x - \sin^2 x - 1 = 0\) and then \( \sin^2 x - 3 \sin x + 1 = 0\), which is also correct.
4. The quadratic equation is factored to \((\sin x - 3)(\sin x - 1) = 0\). Here, the factorization is actually incorrect because it would imply that the quadratic expression is factorable into those terms, which does not hold for this case since the roots do not satisfy the equation.
- The roots derived from the factors suggest:
- \(\sin x = 3\), which is not possible since the sine function ranges from -1 to 1.
- \(\sin x = 1\), which provides valid solutions.
The error in this transformation occurs primarily in the factored form; hence the conclusion that "the equation was factored incorrectly" is the most accurate description of the error present in the solution.
So, the best response is:
**The equation was factored incorrectly.**
1. The original equation is \(3 \sin x + \cos^2 x = 2\).
2. The transformation using the Pythagorean identity \(\cos^2 x = 1 - \sin^2 x\) is correctly applied to yield \(3 \sin x + (2 - \sin^2 x) = 2\).
3. Rearranging leads to the equation \(3 \sin x - \sin^2 x - 1 = 0\) and then \( \sin^2 x - 3 \sin x + 1 = 0\), which is also correct.
4. The quadratic equation is factored to \((\sin x - 3)(\sin x - 1) = 0\). Here, the factorization is actually incorrect because it would imply that the quadratic expression is factorable into those terms, which does not hold for this case since the roots do not satisfy the equation.
- The roots derived from the factors suggest:
- \(\sin x = 3\), which is not possible since the sine function ranges from -1 to 1.
- \(\sin x = 1\), which provides valid solutions.
The error in this transformation occurs primarily in the factored form; hence the conclusion that "the equation was factored incorrectly" is the most accurate description of the error present in the solution.
So, the best response is:
**The equation was factored incorrectly.**