SOHCAHTOA Ladder Quiz: Ladder Against Wall Scenario (SOHCAHTOA Method)

The answer is True. sin(0 degrees) = 0, so height = ladder times 0 = 0 m regardless of the ladder length. A 0 degree angle means the ladder is horizontal, lying flat along the ground parallel to it. In this position the top of the ladder is at ground level and therefore touches no part of the wall above the floor. This represents the minimum height case of the ladder problem.

Height = ladder times sin(angle). Option A: 10 times sin(60 degrees) = 10 times 0.866 = 8.66 m, which is correct. Option B: 15 times sin(30 degrees) = 15 times 0.5 = 7.5 m, not 8.66 m. Option C: 20 times sin(30 degrees) = 20 times 0.5 = 10 m, not 8.66 m. Option D: 10 times sin(30 degrees) = 10 times 0.5 = 5 m, not 8.66 m. Only option A produces the required height.

sin(theta) = 12 divided by 13 = 0.9231. Applying the inverse sine gives theta = arcsin(0.9231) = 67.38 degrees, which rounds to 67.4 degrees. This corresponds to the 5-12-13 right triangle, a standard Pythagorean triple. Option B gives 70 degrees where sin(70 degrees) = 0.940, which would produce a height of 12.22 ft for a 13 ft ladder. Options C and D give even larger discrepancies from the calculated sine value.

Sine increases steadily from 0 to 1 as the angle increases from 0 to 90 degrees. sin(0 degrees) = 0, sin(30 degrees) = 0.5, sin(60 degrees) = 0.866, and sin(90 degrees) = 1. Because height = ladder times sin(theta), the height increases alongside the sine value. Option A describes the behavior of cosine, not sine. Option B is incorrect because sine changes continuously with angle. Option D describes the behavior of sine beyond 90 degrees, not within the 0 to 90 range.

Height = 10 times sin(0 degrees) = 10 times 0 = 0 m. At 0 degrees the ladder lies flat on the ground, so its tip is at ground level and reaches no height on the wall. Option B suggests half the ladder length, which would require sin(theta) = 0.5, corresponding to 30 degrees. Option C would mean the full ladder length is reached vertically, which requires a 90 degree angle. Option D is incorrect because the formula is perfectly applicable and gives a definite answer of 0 m.

The answer is True. Height = ladder times sin(theta). When the angle is fixed, sin(theta) is a constant multiplier. A longer ladder multiplied by that same constant produces a proportionally greater height. For example, at 60 degrees, a 10 m ladder reaches 8.66 m while a 20 m ladder reaches 17.32 m. The height scales directly and proportionally with the ladder length whenever the angle remains the same.

Height = 10 times sin(75 degrees) = 10 times 0.9659 = 9.659 m, which rounds to 9.66 m. sin(75 degrees) is approximately 0.966, a value close to 1 because 75 degrees is close to 90 degrees. Option B gives 8.5 m, which would require sin(theta) = 0.85, corresponding to approximately 58.2 degrees. Option C gives 7.5 m, corresponding to sin(theta) = 0.75 and an angle of 48.6 degrees. Option D gives 9 m, requiring sin(theta) = 0.9 and an angle of about 64 degrees.

The ladder is the hypotenuse of the right triangle and the wall height is the opposite side. Since sin(theta) = opposite divided by hypotenuse, rearranging gives height = hypotenuse times sin(theta). Options A and D state this same correct formula in two equivalent ways since the ladder and hypotenuse refer to the same side. Option B uses cosine, which gives the horizontal base distance not the height. Option C divides by sin(theta) instead of multiplying, which inverts the correct relationship.

sin(theta) = height divided by ladder = 5 divided by 10 = 0.5. Applying the inverse sine gives theta = arcsin(0.5) = 30 degrees. sin(30 degrees) = 0.5 is a standard exact value. Option B gives sin(45 degrees) = 0.707, producing a height of 7.07 m for a 10 m ladder. Option C gives sin(60 degrees) = 0.866, producing 8.66 m. Option D gives sin(25 degrees) = 0.423, producing 4.23 m, not 5 m.

Height = 8 times sin(37 degrees) = 8 times 0.6018 = 4.81 m, which rounds to 5 m. sin(37 degrees) approximately equals 0.6018, which is close to 0.6, matching a scaled 3-4-5 triangle. Option A gives 4.5 m, which would require sin(theta) = 0.5625, corresponding to approximately 34.2 degrees. Option C gives 6 m, requiring sin(theta) = 0.75, corresponding to 48.6 degrees. Option D gives 7 m, requiring sin(theta) = 0.875, corresponding to about 61 degrees.

The ladder is the hypotenuse and the height is the opposite side, so height = ladder times sin(angle) = 10 times sin(60 degrees) = 10 times 0.866 = 8.66 m. Option B uses cos(60 degrees) instead of sine, giving the horizontal distance. Option C corresponds to sin(48.6 degrees), not 60 degrees. Option D does not correspond to any standard trigonometric result for this triangle.

Height = ladder times sin(theta). Since sine increases as the angle increases from 0 to 90 degrees, the height reached also increases. At 0 degrees the height is zero, and at 90 degrees the height equals the full ladder length. This means tilting the ladder more steeply toward the wall always raises the point where it touches, as long as the angle stays between 0 and 90 degrees.

Height = 25 times sin(53 degrees) = 25 times 0.7986 = 19.97 m, which rounds to 20 m. This is consistent with a scaled 3-4-5 triangle where sin(53 degrees) approximately equals 4/5. Option B would require sin(theta) = 15/25 = 0.6, corresponding to 36.87 degrees, not 53. Option C would mean the ladder is fully vertical. Option D gives 21 m, which would require sin(theta) = 0.84, corresponding to approximately 57 degrees, not 53.

Rearranging sin(theta) = height divided by ladder, multiplying both sides by ladder gives height = ladder times sin(theta). This is the direct application of the sine ratio where the ladder is the hypotenuse and the wall height is the opposite side. Option A inverts the relationship. Option B places sin(theta) in the numerator when it should be a multiplier. Option C uses cosine, which gives the horizontal ground distance from the wall to the ladder base, not the height.

sin(theta) = height divided by ladder = 17.32 divided by 20 = 0.866. Applying the inverse sine gives theta = arcsin(0.866) = 60 degrees. sin(60 degrees) = sqrt(3)/2 approximately equals 0.866, confirming the result. Option B gives sin(45 degrees) = 0.707, which would produce a height of 14.14 ft. Option C gives sin(30 degrees) = 0.5, producing a height of 10 ft. Option D gives sin(75 degrees) = 0.966, producing a height of 19.32 ft.

The answer is False. Height = ladder times sin(theta). As the angle decreases toward 0 degrees, sin(theta) also decreases toward 0, so the height decreases. A smaller angle means the ladder is more tilted toward the ground, bringing the top of the ladder lower on the wall. The greatest height is achieved at 90 degrees when sin(theta) = 1. Increasing the angle from 0 to 90 degrees steadily increases the height reached.

Height = 15 times sin(30 degrees) = 15 times 0.5 = 7.5 ft. sin(30 degrees) = 0.5 is a standard value. Option B would require sin(theta) = 10/15 = 0.667, corresponding to approximately 41.8 degrees, not 30. Option C would require sin(theta) = 12/15 = 0.8, corresponding to 53.1 degrees. Option D would mean the ladder is vertical, which requires a 90 degree angle, not 30.

The ladder forms the hypotenuse of a right triangle, and the height reached is the side opposite the angle. Since sine equals opposite divided by hypotenuse, rearranging gives height = hypotenuse times sine of the angle = ladder times sin(theta). Option A uses cosine, which gives the horizontal distance from the wall to the base of the ladder, not the height. Option B uses tangent, which relates opposite to adjacent and is not directly applicable here. Option D is the reciprocal of tangent.

The answer is True. sin(90 degrees) = 1, which is the maximum value of sine. Using height = ladder times sin(theta), when theta = 90 degrees the height equals the full length of the ladder. The ladder stands perfectly upright against the wall, and no other angle produces a greater height for the same ladder length. In practice a vertical ladder would not lean against a wall, but mathematically this represents the maximum height case.

Height = 12 times sin(45 degrees) = 12 times 0.7071 = 8.485 m, which rounds to 8.49 m. Option A rounds incorrectly to 8.5 m, which is not precise enough to match the exact calculation. Option B does not correspond to sin(45 degrees) for a 12 m ladder. Option D would require sin(theta) = 10/12 = 0.833, which corresponds to approximately 56.4 degrees, not 45 degrees.