Question 1: In a right triangle, the length of the adjacent side is 8 cm and the hypotenuse is 10 cm. Calculate the value of sin θ using CAH and identify any mistake if you find an incorrect approach.

Answer: sin θ = 8/10 = 0.8

Question 2: A student computes cos θ as 8/10 and then concludes that the opposite side is 6 cm. Explain what is wrong with this reasoning and correct it.

Answer: They used cos θ=adj/hyp but then incorrectly found opposite side. Correct calculation: opposite = hyp * sin θ, or find sin θ first and then use Pythagoras.

Question 3: Construct a right triangle where the cosine of angle θ is 0.6, and the hypotenuse is 10 cm. Show your calculations and label the sides.

Answer: adjacent = 10 * 0.6 = 6 cm; opposite = √(10² - 6²) = 8 cm.

Question 4: Identify the common mistake in calculating CAH for a triangle with an adjacent side of 5 cm and hypotenuse of 13 cm, where a student claims cos θ = 13/5.

Answer: They inverted the ratio. Correct: cos θ = adjacent/hypotenuse = 5/13.

Question 5: A right triangle has an angle θ where the opposite side is 7 cm, and the hypotenuse is 14 cm. Calculate cos θ and explain if the student’s mistake in using CAH occurs if they used 7/14 directly for cos.

Answer: cos θ = √(hyp² - opposite²) / hyp = √(14² - 7²)/14 = √(196 - 49)/14 = √147/14. Student mistake: using 7/14 as cos θ, which is actually sin θ.

Question 6: A problem involves a ladder leaning against a wall, forming a right triangle with the ground. If the ladder makes an angle where CAH indicates cos θ relates to the distance from wall and ladder length, identify a common misconception students may have.

Answer: Students may confuse adjacent and hypotenuse or forget to check if ratios are correctly applied to the sides.

Question 7: Calculate the missing side in a triangle where cos θ = 0.5 and hypotenuse = 10 cm. Show your work and explain the common error students make with CAH in similar problems.

Answer: adjacent = 10 * 0.5 = 5 cm; opposite = √(10² - 5²) = √75 ≈ 8.66 cm. Common error: forgetting to use Pythagoras after calculating cosine.

Question 8: Given a triangle with adjacent side 9 cm and hypotenuse 15 cm, find sin θ and identify what students often overlook when applying CAH ratios.

Answer: sin θ = √(hyp² - adjacent²)/hyp = √(15² - 9²)/15 = √(225 - 81)/15 = √144/15 = 12/15 = 0.8. Overlook: not verifying that ratios are correct or misidentifying sides.

Question 9: A student claims that using CAH ratios always guarantees correct side calculations. Write a short explanation of a scenario where this might lead to an error, and how to avoid it.

Answer: Incorrect: assuming the triangle is right-angled or mislabeling sides. Always verify sides and angles before applying ratios.

Question 10: Challenge question: In a right triangle, if cos θ = 0.8, find both sin θ and tan θ. Explain if students often make errors in these calculations.

Answer: sin θ = √(1 - 0.8²) = √(1 - 0.64) = √0.36 = 0.6; tan θ = sin θ / cos θ = 0.6 / 0.8 = 0.75. Common error: confusing the ratios or mixing up sin and cos.