The question requires finding cos(θ) in a right triangle where the side adjacent to acute angle θ is 9 and the hypotenuse is 15. According to SOH-CAH-TOA, cosine is the ratio of adjacent over hypotenuse, so cos(θ) = adjacent / hypotenuse. Set up the ratio as cos(θ) = 9 / 15. Simplify 9/15 to 3/5. A key error to avoid is mixing up adjacent and opposite sides, which might lead to using sine instead. For test-taking, double-check the side identifications relative to the given angle before selecting the ratio.

This asks for the hypotenuse in right triangle MNO with right angle at O, angle M at 48 degrees, and opposite side 17, to the nearest tenth. Sine is opp/hyp from SOH-CAH-TOA. Set up sin(48°) = 17 / hypotenuse. Solve hypotenuse = 17 / sin(48°) ≈ 17 / 0.7431 ≈ 22.87, rounding to 22.9. Using cosine mistakenly requires adjacent side. Ensure precise sine values for accurate rounding.

The question requires the measure of angle A at (0,0) in right triangle ABC with points B(6,0), C(6,8), right at B, to the nearest degree. Tangent applies with opposite BC=8 and adjacent AB=6 relative to A via SOH-CAH-TOA. Set up tan(A) = 8 / 6 ≈ 1.333. Calculate A = tan⁻¹(1.333) ≈ 53.13 degrees, rounding to 53 degrees. Confusing coordinates might lead to wrong sides, but plot points to confirm. Verify with other ratios like sine for consistency.

This problem asks for acute angle θ to the nearest degree in a right triangle with opposite side 16 and adjacent side 30. Tangent is opposite over adjacent per SOH-CAH-TOA. Set up tan(θ) = 16 / 30 ≈ 0.5333. Calculate θ = tan⁻¹(0.5333) ≈ 28.07 degrees, rounding to 28 degrees. A mistake is using hypotenuse, but it's not given. Calculate hypotenuse to check if ratios match.

The question seeks the measure of angle J in right triangle JKL with right angle at K, JK = 14, and hypotenuse JL = 20, to the nearest degree. Cosine is adj/hyp from SOH-CAH-TOA, with JK adjacent to J. Set up cos(J) = 14 / 20 = 0.7. Then, J = cos⁻¹(0.7) ≈ 45.57 degrees, rounding to 46 degrees. Errors occur if opposite is used instead of adjacent. Compute the missing side to verify consistency.

The question finds the height of a drone after ascending 120 meters at a 25-degree angle with the ground, to the nearest meter. Sine is appropriate as height is opposite the angle and path is hypotenuse via SOH-CAH-TOA. Set up sin(25°) = height / 120. Calculate height = 120 × sin(25°) ≈ 120 × 0.4226 = 50.712, rounding to 51 meters. Using tangent incorrectly assumes horizontal distance given. Verify the given length is hypotenuse for sine.

This asks for EF in right triangle DEF with right angle at F, angle D at 62 degrees, and adjacent DF = 9, to the nearest tenth. Tangent fits as EF is opposite and DF adjacent to D per SOH-CAH-TOA. Set up tan(62°) = EF / 9. Calculate EF = 9 × tan(62°) ≈ 9 × 1.8807 = 16.926, rounding to 16.9. Confusing with cosine would find hypotenuse instead. Sketch to confirm opposite vs. adjacent.

The problem requires tan(θ) in a right triangle with legs 8 and 15, where θ is adjacent to 15 and opposite 8. Tangent is opposite over adjacent from SOH-CAH-TOA. Set up tan(θ) = 8 / 15. The hypotenuse is sqrt(8² + 15²) = 17, but unnecessary here. Errors include using hypotenuse, giving sin or cos instead. Identify sides precisely relative to θ before ratio selection.

This question finds the height of a flagpole with a 54-degree angle of elevation from 18 meters away, to the nearest tenth. Tangent applies as height is opposite and distance adjacent to the angle per SOH-CAH-TOA. Set up tan(54°) = height / 18. Calculate height = 18 × tan(54°) ≈ 18 × 1.3764 = 24.775, rounding to 24.8 meters. Using sine mistakenly treats distance as hypotenuse. Confirm no hypotenuse is involved for tangent problems.

The problem seeks cos(φ) given sin(θ) = 0.6, where θ and φ are complementary acute angles in a right triangle. Since φ = 90° - θ, cos(φ) = sin(θ) by cofunction identity. Thus, cos(φ) = 0.6. Using SOH-CAH-TOA directly, sin(θ) means opposite/hypotenuse for θ is adjacent/hypotenuse for φ. A common error is thinking cos(φ) = cos(θ), but remember complementary relationships. This highlights using identities for related angles.