Question

level 4: using full sentences, answer the questions below!
in the right triangle below, what ratio(s) is equal to \\(\frac{8}{17}\\)? how do you know?
triangle with sides ab = 8, bc = 15, ac = 17

in the right triangle below, which equation is not correct? how do you know?
triangle with sides ac = 12, bc = 16, ab = 20
a. \\(\cos a = \frac{12}{20}\\)
b. \\(\tan a = \frac{16}{12}\\)
c. \\(\sin b = \frac{12}{20}\\)
d. \\(\tan b = \frac{26}{20}\\)

level 5: error analysis! both of the problems below are incorrect. write a sentence, or two, describing the error in the work, and write the correct trig ratio.
triangle with sides ac = 5, bc = 12, ab = 13, work shown: \\(\sin a = \frac{5}{13}\\)
triangle with sides de = 12, ef = 35, df = 37, work shown: \\(\tan d = \frac{35}{37}\\)

Explanation:

Orient the upside-down image and identify questions

The image is upside down. Rotating it \(180^\circ\) reveals four distinct problems divided into two levels:

• Level 4, Question 1: In the right triangle below, what ratio(s) is equal to \(\frac{8}{17}\)? How do you know?
• Triangle \(ABC\) has right angle at \(B\), \(AB = 8\), \(BC = 15\), and hypotenuse \(AC = 17\).
• Level 4, Question 2: In the right triangle below, which equation is NOT correct? How do you know?
• Triangle \(ABC\) has right angle at \(C\), \(AC = 12\), \(BC = 16\), and hypotenuse \(AB = 20\).
• Options: A. \(\cos A = \frac{12}{20}\), B. \(\tan A = \frac{16}{12}\), C. \(\sin B = \frac{12}{20}\), D. \(\tan B = \frac{26}{20}\).
• Level 5, Question 3 (Error Analysis 1): Describe the error in the work and write the correct trig ratio for:
• Right triangle \(ABC\) with right angle at \(C\), \(AC = 5\), \(BC = 12\), \(AB = 13\).
• Given work: \(\sin A = \frac{5}{13}\).
• Level 5, Question 4 (Error Analysis 2): Describe the error in the work and write the correct trig ratio for:
• Right triangle \(DEF\) with right angle at \(E\), \(DE = 12\), \(EF = 35\), \(DF = 37\).
• Given work: \(\tan D = \frac{35}{37}\).

Solve Level 4 Questions

• Question 1:
• \(\text{Opposite to } A = 15\), \(\text{Adjacent to } A = 8\), \(\text{Hypotenuse} = 17\).
• \(\text{Opposite to } C = 8\), \(\text{Adjacent to } C = 15\), \(\text{Hypotenuse} = 17\).
• \(\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8}{17}\).
• \(\sin C = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{8}{17}\).
• Question 2:
• \(\cos A = \frac{AC}{AB} = \frac{12}{20}\) (Correct)
• \(\tan A = \frac{BC}{AC} = \frac{16}{12}\) (Correct)
• \(\sin B = \frac{AC}{AB} = \frac{12}{20}\) (Correct)
• \(\tan B = \frac{AC}{BC} = \frac{12}{16}\). Option D states \(\tan B = \frac{26}{20}\), which is incorrect.

Solve Level 5 Questions

• Question 3:
• For angle \(A\), \(\text{Opposite} = BC = 12\), \(\text{Hypotenuse} = AB = 13\).
• Correct ratio: \(\sin A = \frac{12}{13}\).
• Error: The work used the adjacent side (\(AC = 5\)) instead of the opposite side (\(BC = 12\)) in the numerator.
• Question 4:
• For angle \(D\), \(\text{Opposite} = EF = 35\), \(\text{Adjacent} = DE = 12\).
• Correct ratio: \(\tan D = \frac{35}{12}\).
• Error: The work used the hypotenuse (\(DF = 37\)) instead of the adjacent side (\(DE = 12\)) in the denominator.

• \(\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{8}{17}\)
• \(\sin C = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{8}{17}\)

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Question 2

The incorrect equation is D. \(\tan B = \frac{26}{20}\).

The tangent of an angle is defined as \(\frac{\text{Opposite}}{\text{Adjacent}}\). For angle \(B\), the opposite side is \(AC = 12\) and the adjacent side is \(BC = 16\), so the correct ratio is \(\tan B = \frac{12}{16}\). Option D incorrectly uses values that do not correspond to these sides.

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Level 5: Error Analysis

Question 3

• Error Description: The student used the adjacent side (\(AC = 5\)) instead of the opposite side (\(BC = 12\)) in the numerator for the sine ratio.
• Correct Ratio: \(\sin A = \frac{12}{13}\)

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Question 4

• Error Description: The student used the hypotenuse (\(DF = 37\)) instead of the adjacent side (\(DE = 12\)) in the denominator for the tangent ratio.
• Correct Ratio: \(\tan D = \frac{35}{12}\)

</answer>

Answer:

Level 4

Question 1

The ratios equal to \(\frac{8}{17}\) are \(\cos A\) and \(\sin C\).

Explanation:

• \(\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{8}{17}\)
• \(\sin C = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{8}{17}\)

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Question 2

The incorrect equation is D. \(\tan B = \frac{26}{20}\).

Explanation:
The tangent of an angle is defined as \(\frac{\text{Opposite}}{\text{Adjacent}}\). For angle \(B\), the opposite side is \(AC = 12\) and the adjacent side is \(BC = 16\), so the correct ratio is \(\tan B = \frac{12}{16}\). Option D incorrectly uses values that do not correspond to these sides.

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Level 5: Error Analysis

Question 3

• Error Description: The student used the adjacent side (\(AC = 5\)) instead of the opposite side (\(BC = 12\)) in the numerator for the sine ratio.
• Correct Ratio: \(\sin A = \frac{12}{13}\)

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Question 4

• Error Description: The student used the hypotenuse (\(DF = 37\)) instead of the adjacent side (\(DE = 12\)) in the denominator for the tangent ratio.
• Correct Ratio: \(\tan D = \frac{35}{12}\)