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Sagot :
Let's analyze each step in the solution provided to determine whose procedure is correct.
1. Step 1 Analysis:
Given equation:
[tex]\[ \frac{\left(-\frac{8}{17}\right)^2}{\cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta} \][/tex]
Rewrite the equation:
[tex]\[ \frac{\left(-\frac{8}{17}\right)^2}{\cos^2 \theta} + 1 = \sec^2 \theta \][/tex]
Since [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex], this implies:
[tex]\[ \sec^2 \theta = 1 + \tan^2 \theta \][/tex]
2. Step 2 Analysis:
Simplifying the left side of the equation:
[tex]\[ \frac{\left(-\frac{8}{17}\right)^2}{\cos^2 \theta} + 1 \][/tex]
Calculate [tex]\(\left(-\frac{8}{17}\right)^2\)[/tex]:
[tex]\[ \left(-\frac{8}{17}\right)^2 = \frac{64}{289} \][/tex]
Now, the equation becomes:
[tex]\[ \frac{\frac{64}{289}}{\cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta} \][/tex]
3. Step 3 Analysis:
Multiply everything by [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \frac{64}{289} + \cos^2 \theta = 1 \][/tex]
4. Step 4 Analysis:
Rearrange the equation to solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \frac{64}{289} \][/tex]
5. Step 5 Analysis:
Simplify the right side:
[tex]\[ \cos^2 \theta = \frac{289}{289} - \frac{64}{289} \][/tex]
[tex]\[ \cos^2 \theta = \frac{225}{289} \][/tex]
6. Step 6 Analysis:
Take the square root of both sides to solve for [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos \theta = \pm \sqrt{\frac{225}{289}} \][/tex]
[tex]\[ \cos \theta = \pm \frac{15}{17} \][/tex]
Given the detailed, step-by-step solution, we see that:
- Each step logically follows from the previous one and correctly applies algebraic and trigonometric rules.
- Therefore, the provided solution is correct.
Comparing this to the given answers:
- Keisha's procedure is correct.
- David's procedure is correct.
- Both procedures are correct.
- Neither procedure is correct.
Since the detailed solution is correct based on the rules and calculations, it implies that both procedures are correct when followed properly. Therefore, the correct response is:
Both procedures are correct.
1. Step 1 Analysis:
Given equation:
[tex]\[ \frac{\left(-\frac{8}{17}\right)^2}{\cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta} \][/tex]
Rewrite the equation:
[tex]\[ \frac{\left(-\frac{8}{17}\right)^2}{\cos^2 \theta} + 1 = \sec^2 \theta \][/tex]
Since [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex], this implies:
[tex]\[ \sec^2 \theta = 1 + \tan^2 \theta \][/tex]
2. Step 2 Analysis:
Simplifying the left side of the equation:
[tex]\[ \frac{\left(-\frac{8}{17}\right)^2}{\cos^2 \theta} + 1 \][/tex]
Calculate [tex]\(\left(-\frac{8}{17}\right)^2\)[/tex]:
[tex]\[ \left(-\frac{8}{17}\right)^2 = \frac{64}{289} \][/tex]
Now, the equation becomes:
[tex]\[ \frac{\frac{64}{289}}{\cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta} \][/tex]
3. Step 3 Analysis:
Multiply everything by [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \frac{64}{289} + \cos^2 \theta = 1 \][/tex]
4. Step 4 Analysis:
Rearrange the equation to solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \frac{64}{289} \][/tex]
5. Step 5 Analysis:
Simplify the right side:
[tex]\[ \cos^2 \theta = \frac{289}{289} - \frac{64}{289} \][/tex]
[tex]\[ \cos^2 \theta = \frac{225}{289} \][/tex]
6. Step 6 Analysis:
Take the square root of both sides to solve for [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos \theta = \pm \sqrt{\frac{225}{289}} \][/tex]
[tex]\[ \cos \theta = \pm \frac{15}{17} \][/tex]
Given the detailed, step-by-step solution, we see that:
- Each step logically follows from the previous one and correctly applies algebraic and trigonometric rules.
- Therefore, the provided solution is correct.
Comparing this to the given answers:
- Keisha's procedure is correct.
- David's procedure is correct.
- Both procedures are correct.
- Neither procedure is correct.
Since the detailed solution is correct based on the rules and calculations, it implies that both procedures are correct when followed properly. Therefore, the correct response is:
Both procedures are correct.
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