Sun Angle Shadow Quiz: Sun Angle From Height and Shadow

θ = arctan(2.4/3.1) = arctan(0.7742) = 37.74°. Among the options, 37.9° is closest to 37.74°. Option A gives 37.5°, which is 0.24° away. Option B gives 37.9°, which is only 0.16° away — the closest. Option C gives 38.3°, which is 0.56° away. Option D gives 38.7°, which is 0.96° away. Only 37.9° minimizes the distance to the computed value of 37.74°.

θ = arctan(13.2/29.7) = arctan(0.4444) = 23.96°. Option B gives 21.50°, where tan21.50° = 0.3939, giving height = 29.7×0.3939 = 11.70 m, not 13.2 m. Option C gives 25.80°, where tan25.80° = 0.4841, giving 14.38 m. Option D gives 27.40°, where tan27.40° = 0.5184, giving 15.40 m. Only 23.96° satisfies arctan(0.4444).

θ = arctan(9.0/4.0) = arctan(2.25) = 66.04°, which rounds to 66°. Option A gives 65°, where tan65° = 2.145, giving shadow = 9.0/2.145 = 4.20 m, not 4.0 m. Option C gives 67°, where tan67° = 2.356, giving 3.82 m. Option D gives 68°, where tan68° = 2.475, giving 3.64 m. Only 66° satisfies the given ratio to the nearest degree.

θ = arctan(5/5) = arctan(1) = 45°. When height equals shadow, the ratio is 1 and arctan(1) = 45° exactly. Option A gives tan30° = 1/√3 ≈ 0.577, meaning height would be about 0.577 times the shadow. Option B gives tan60° = √3 ≈ 1.73, meaning height would be √3 times the shadow. Option D gives 90°, which would require an infinitely long or zero-length shadow.

The answer is True. θ = arctan(height/shadow). Doubling both gives arctan(2×height/2×shadow) = arctan(height/shadow) = θ. The factor of 2 cancels in the ratio. This confirms that θ depends only on the proportion between height and shadow, not their absolute values. A 3 m pole with a 6 m shadow gives the same angle as a 6 m pole with a 12 m shadow, since both produce tanθ = 0.5.

For positive height and shadow, tanθ = height/shadow > 0, placing θ strictly between 0° and 90°, confirming A. Very long shadows make the ratio small, giving θ close to 0°, confirming B. Very short shadows make the ratio large, giving θ close to 90°, confirming C. Option D is false — when shadow > height, tanθ < 1 means θ < 45°, still within 0° to 90°.

θ = arctan(18.5/25.0) = arctan(0.74) = 36.47° ≈ 36.50°. Option B gives 34.20°, where tan34.20° = 0.682, giving height = 25×0.682 = 17.05 m, not 18.5 m. Option C gives 38.80°, where tan38.80° = 0.803, giving 20.08 m. Option D gives 40.10°, where tan40.10° = 0.843, giving 21.08 m. Only 36.50° satisfies arctan(0.74).

θ = arctan(7.2/12.0) = arctan(0.6) = 30.964° ≈ 31.0°. Option A gives 32.0°, where tan32° = 0.6249, giving height = 12×0.6249 = 7.50 m, not 7.2 m. Option C gives 30.0°, where tan30° = 0.5774, giving 6.93 m. Option D gives 33.0°, where tan33° = 0.6494, giving 7.79 m. Only 31.0° satisfies the given ratio to 1 decimal place.

θ = arctan(3.75/8.25) = arctan(0.4545) = 24.44°. Option B gives 26.10°, where tan26.10° = 0.490, giving height = 8.25×0.490 = 4.04 m, not 3.75 m. Option C gives 22.80°, where tan22.80° = 0.420, giving 3.47 m. Option D gives 28.50°, where tan28.50° = 0.543, giving 4.48 m. Only 24.44° satisfies arctan(0.4545).

θ = arctan(6.0/10.0) = arctan(0.6) = 30.964° ≈ 30.96°. Option B gives 32.96°, where tan32.96° = 0.6472, giving height = 10×0.6472 = 6.47 m, not 6.0 m. Option C gives 28.96°, where tan28.96° = 0.5370, giving 5.37 m. Option D gives 35.96°, where tan35.96° = 0.7239, giving 7.24 m. Only 30.96° satisfies the given ratio of 0.6.

tanθ = height/shadow = √3×shadow/shadow = √3. Since tan60° = √3, θ = 60°. Option A gives tan30° = 1/√3 ≈ 0.577, which would mean the height is 1/√3 times the shadow, not √3 times. Option B gives tan45° = 1, meaning equal height and shadow. Option D gives tan75° ≈ 3.73, which is much greater than √3 ≈ 1.73.

Options A and B are equivalent correct formulations using the right-triangle tangent definition. Option D is valid because the k cancels in the ratio, giving arctan(height/shadow). Option C is wrong because arcsin requires the ratio of opposite to hypotenuse, not opposite to adjacent — height/shadow is not height/hypotenuse. 

The answer is True. tanθ = height/shadow. If height is in centimeters and shadow is in meters, the ratio becomes (height in cm)/(shadow in m). Since 1 m = 100 cm, this is equivalent to computing 100 times the correct ratio. For example, height = 200 cm and shadow = 3 m gives ratio = 200/3 = 66.7 instead of the correct 2/3 = 0.667, producing θ = arctan(66.7) ≈ 89.1° instead of the correct 33.7°.

Option A inverts the ratio, computing the complement of the correct angle instead. Option B causes the calculator to misinterpret the angle unit, producing a completely wrong result. Option C changes the ratio since mixing centimeters and meters without conversion scales one measurement by 100 relative to the other. Option D violates the flat-ground assumption on which the right triangle model is built.

The answer is True. The shadow is a physical length measured along the ground from the base of the object. Length is always a non-negative quantity. In the right triangle model, the shadow is the adjacent side, which represents a distance and cannot be negative. Entering a negative shadow value into the formula would produce a negative ratio and a negative or undefined angle, neither of which has meaning in this context.

When height increases with shadow fixed, the ratio height/shadow increases, so arctan of a larger value gives a larger θ, confirming A. When shadow increases with height fixed, the ratio decreases, giving a smaller θ, confirming B. θ is the acute angle between the horizontal ground and the incoming sunlight direction, confirming C. Option D is false — using inconsistent units such as centimeters for height and meters for shadow without converting changes the ratio and therefore changes θ.

The answer is True. As shadow length approaches 0 from above, the ratio height/shadow approaches infinity. Since arctan(∞) = 90°, θ approaches 90°. This corresponds physically to the sun being directly overhead, where the shadow vanishes and the line of sunlight is vertical. The model correctly captures this limiting case, though a shadow of exactly zero makes the ratio undefined as a strict mathematical limit.

tan40° = height/shadow, so shadow = height/tan40° = 5.0/0.8391 = 5.96 m. Option B gives 4.20, requiring tan(θ) = 5.0/4.20 = 1.190, corresponding to approximately 50°. Option C gives 7.15, requiring tan(θ) = 0.699, corresponding to approximately 34.9°. Option D gives 3.80, requiring tan(θ) = 1.316, corresponding to approximately 52.7°. Only 5.96 m satisfies the equation with θ = 40°.

Option A is the direct definition, confirming A. arccot(s/h) = arctan(h/s) since arccot(x) = arctan(1/x), confirming B. Multiplying both h and s by k cancels in the ratio leaving arctan(h/s) unchanged, confirming C. Option D states θ = arcsin(h/s), which is false — arcsin requires the ratio of opposite to hypotenuse, not opposite to adjacent. The hypotenuse is √(h²+s²), not s, so arcsin(h/s) does not give the correct angle.

The answer is True. The vertical object forms the opposite side and the shadow forms the adjacent side of a right triangle. By definition, tan(θ) = opposite/adjacent = height/shadow. Applying the inverse tangent gives θ = arctan(height/shadow). This is the standard right-triangle model used whenever the ground is level and the object is perfectly vertical.