Q: Simplify the trigonometric expression (0° < A < 90°): Evaluate sin A / (1 + cos A) + sin A / (1 - cos A) and choose the correct simplified form.

Introduction / Context: This item asks for algebraic simplification of a standard trigonometric sum involving sin A and cos A. The interval ensures all expressions are defined (excluding division by zero cases). Given Data / Assumptions: Expression: sin A/(1 + cos A) + sin A/(1 - cos A). Angle A in (0°, 90°). Concept / Approach: Put both terms over a common denominator, use 1 - cos^2 A = sin^2 A, and simplify carefully. Step-by-Step Solution: Common denominator = (1 + cos A)(1 - cos A) = 1 - cos^2 A = sin^2 A.Numerator = sin A (1 - cos A) + sin A (1 + cos A) = sin A (2) = 2 sin A.Thus value = (2 sin A) / (sin^2 A) = 2 / sin A = 2 cosec A. Verification / Alternative check: Pick A = 30°: LHS = (0.5/1.866...) + (0.5/0.134..) ≈ 0.268 + 3.732 ≈ 4.0; RHS = 2 * cosec 30° = 2 * 2 = 4 ✔️. Why Other Options Are Wrong: 2 sec A, 2 sin A, 2 cos A mismatch after the Pythagorean substitution and do not hold numerically for test angles. Common Pitfalls: Forgetting the identity 1 - cos^2 A = sin^2 A; sign errors when expanding; canceling incorrectly before combining fractions. Final Answer: 2 cosec A

Q: Leaning ladder against a wall: A ladder makes a 60° angle with the horizontal ground, and its foot is 4.6 m away from the wall. Find the length of the ladder.

Introduction / Context: A ladder against a wall forms a right triangle with the ground (adjacent) and wall (opposite). The angle with ground and the horizontal distance to the wall are known; find the hypotenuse (ladder length). Given Data / Assumptions: Angle with ground = 60°. Adjacency (foot to wall) = 4.6 m. Ladder is straight; wall and ground are perpendicular. Concept / Approach: Use cos θ = adjacent / hypotenuse → hypotenuse = adjacent / cos θ. Step-by-Step Solution: cos 60° = 1/2.Length L = 4.6 / (1/2) = 9.2 m. Verification / Alternative check: Compute vertical reach: L * sin 60° = 9.2 * (√3/2) ≈ 7.97 m; Pythagoras with 4.6 m confirms hypotenuse ≈ 9.2 m. Why Other Options Are Wrong: 2.3 and 4.6 m are too short; 7.8 m corresponds to cos near 0.59, not 0.5. Common Pitfalls: Using tan instead of cos; mixing degrees with radians; rounding too early on √3 or cos values. Final Answer: 9.2 m

Q: Depression to tower top and bottom from a cliff: From the top of a cliff of height h metres, the angles of depression of the top and the bottom of a nearby vertical tower are 30° and 60° respectively. Find the height of the tower in terms of h.

Introduction / Context: Two angles of depression from a higher point to two points on or above the ground allow expressing horizontal distance and relative verticals, yielding the unknown tower height as a function of h. Given Data / Assumptions: Cliff height = h. Angles of depression: 30° to tower top, 60° to tower bottom. Tower is vertical; ground is level. Concept / Approach: If horizontal distance from cliff foot to tower base is x, then tan 60° = h / x → x = h/√3. For the top, vertical drop is h - T (T = tower height). tan 30° = (h - T)/x → solve for T. Step-by-Step Solution: x = h / √3 from tan 60°.tan 30° = (h - T)/x = 1/√3.Thus h - T = x/√3 = (h/√3)/√3 = h/3.So T = h - h/3 = 2h/3. Verification / Alternative check: Insert T back: Top drop = h - 2h/3 = h/3 → tan 30° = (h/3)/(h/√3) = 1/√3 ✔️. Why Other Options Are Wrong: h√3, 2h√3, h/3 mismatch the tangent relations; only 2h/3 satisfies both angles simultaneously. Common Pitfalls: Confusing elevation with depression; misplacing T in h − T; forgetting that both rays share the same horizontal distance x. Final Answer: 2h/3

Question 1

From each vertex A, B, C of a horizontal triangle ABC, the angle of elevation of a hilltop is α (same at all three vertices). If side a is opposite A, find the height of the hill in terms of a and α.

Accepted Answer

Introduction / Context: If the angle of elevation to the same point is equal from all three vertices of triangle ABC, the foot of the perpendicular from the hilltop to the plane lies at the circumcenter O of triangle ABC. The horizontal distance from O to each vertex equals the circumradius R, making the 3D geometry manageable. Given Data / Assumptions: Elevation = α from A, B, C. Let hill height be h; horizontal distance from each vertex to foot O is R (circumradius). Concept / Approach: At any vertex (say A), tan α = h / AO = h / R ⇒ h = R tan α. Triangle relation: a = 2R sin A ⇒ R = a / (2 sin A). Substitute for R. Step-by-Step Solution: R = a / (2 sin A).h = R tan α = (a / (2 sin A)) * tan α = a tan α * csc A / 2. Verification / Alternative check: Using b and B or c and C yields analogous forms (b tan α csc B / 2, etc.). All are consistent because a = 2R sin A, etc. Why Other Options Are Wrong: (a) and (c) omit the division by 2 that comes from R = a/(2 sin A); options mixing sec with A are not from the circumradius identity. Common Pitfalls: Assuming centroid or incenter instead of circumcenter; forgetting that equal elevations imply equal horizontal distances. Final Answer: a tan α cosec A / 2

Question 2

From the bank of a river, the angle subtended by a tree on the opposite bank is 60°. After moving 40 m directly away from the bank, the angle becomes 30°. Find the breadth of the river (in metres).

Accepted Answer

Introduction / Context: This is the same geometry pattern as earlier: two angle observations at known horizontal separations solve for the unknown width. Given Data / Assumptions: Initial angle = 60° at distance x (breadth). After moving 40 m away, angle = 30° at distance x + 40. Concept / Approach: tan 60° = h/x and tan 30° = h/(x + 40) with the same height h; eliminate h to solve for x. Step-by-Step Solution: h = x√3 and h = (x + 40)/√3 ⇒ x√3 = (x + 40)/√3.3x = x + 40 ⇒ 2x = 40 ⇒ x = 20 m. Verification / Alternative check: Check with ratios: (x + 40)/x = √3 * √3 = 3 ⇒ (x + 40) = 3x ⇒ x = 20 m ✔. Why Other Options Are Wrong: 30, 40, 60, 80 m do not keep both tangent equations true simultaneously. Common Pitfalls: Interpreting “angle subtended” as something other than the elevation to the top; mixing up nearer vs farther position. Final Answer: 20 m

Question 3

Angle of elevation changes at the same observation point: From a point 120 m from the base of an unfinished vertical tower, the angle of elevation of the top is 45°. If, after raising the tower, the angle of elevation from the same point must become…

Accepted Answer

Introduction / Context: This problem uses right-triangle trigonometry for angles of elevation from a fixed horizontal distance. The initial angle is 45°; after increasing the tower’s height, the angle becomes 60°. Given Data / Assumptions: Distance from observation point to tower base = 120 m (horizontal). Current angle of elevation = 45°. Target angle of elevation = 60°. Tower is vertical; ground is horizontal; observer’s eye level taken at ground level. Concept / Approach: For a right triangle, tan(theta) = opposite / adjacent. Here, opposite = tower height as seen, adjacent = 120 m. Compute present height, compute required height, take the difference. Step-by-Step Solution: Present height h1 = 120 * tan 45° = 120.Required height h2 = 120 * tan 60° = 120 * √3.Increase = h2 - h1 = 120 (√3 - 1). Verification / Alternative check: If raised by 120 (√3 - 1), the new total becomes 120√3. With adjacent 120, tan = (120√3)/120 = √3 → 60°, consistent. Why Other Options Are Wrong: 120(√3 + 1): adds extra 120 beyond what is needed. 10(√3 + 1): wrong scale; ignores given distance. None of these: incorrect since a correct numeric form exists. Common Pitfalls: Using sine or cosine instead of tangent; forgetting that only the increase is asked (not the final height). Final Answer: 120 ( √3 - 1 ) m

Question 4

Two observations from one line: From a point G on level ground, the angle of elevation of a vertical tower is 30°. After walking 20 m straight towards the tower, the angle of elevation becomes 60°. Find the height of the tower.

Accepted Answer

Introduction / Context: Height from two angles of elevation along the same straight line is a classic tangent setup using two right triangles with a common height and different adjacents. Given Data / Assumptions: Initial angle = 30° at distance x from base. After moving 20 m closer, angle = 60° at distance x - 20. Tower is vertical; ground is horizontal. Concept / Approach: Use tan(theta) = height / horizontal distance for each observation and equate the tower height expressions to solve for x, then compute height. Step-by-Step Solution: Let height = H, initial horizontal = x.tan 30° = H / x → H = x / √3.tan 60° = H / (x - 20) → H = √3 (x - 20).Equate: x / √3 = √3 (x - 20) → x = 3x - 60 → 2x = 60 → x = 30.Thus H = x / √3 = 30 / √3 = 10√3 m. Verification / Alternative check: With H = 10√3 and x = 30, second distance is 10; tan 60° = (10√3)/10 = √3 ✔️. Why Other Options Are Wrong: Values like 20√3 lead to inconsistent distances; only 10√3 satisfies both equations simultaneously. Common Pitfalls: Using 30° with (x - 20) instead of x; mixing up tan and cot; arithmetic slips when isolating x. Final Answer: 10 √3 m

Question 5

Single observation with known height: From point P on level ground, the angle of elevation of the top of a vertical tower is 30°. If the height of the tower is 100 m, find the horizontal distance of P from the foot of the tower.

Accepted Answer

Introduction / Context: Relating a known vertical height and an observed angle of elevation to a horizontal distance uses the tangent definition in right-triangle trigonometry. Given Data / Assumptions: Height H = 100 m. Angle of elevation θ = 30°. Horizontal distance from point to foot = d (unknown). Concept / Approach: tan θ = opposite / adjacent = H / d → solve for d. Step-by-Step Solution: tan 30° = 1 / √3 = 100 / d.So d = 100 √3 m. Verification / Alternative check: Compute tan using result: 100 / (100√3) = 1/√3 → 30°, correct. Why Other Options Are Wrong: 100/√3 would imply tan 30° = √3, which is inverted; 50√3 or 50/√3 do not match the given 100 m height. Common Pitfalls: Confusing tan with cot; swapping opposite and adjacent; rounding √3 too early; answer needs exact form unless asked otherwise. Final Answer: 100 √3 m

Question 6

Angle doubles after moving forward: From a point 160 m from a vertical tower, the angle subtended by the tower is θ. After advancing 100 m straight toward the tower, the subtended angle becomes 2θ. Find the height of the tower.

Accepted Answer

Introduction / Context: Here a single unknown height must satisfy a double-angle relation as the observer moves closer. Using tan identities links the two geometry situations. Given Data / Assumptions: Initial horizontal distance = 160 m; after moving = 60 m. Initial angle = θ; new angle = 2θ. Height of tower = h. Concept / Approach: Set tan θ = h / 160 and tan 2θ = h / 60. Use tan double-angle: tan 2θ = 2 t / (1 - t^2) with t = tan θ. Solve for h. Step-by-Step Solution: Let t = tan θ = h/160.Then tan 2θ = 2t / (1 - t^2) = h/60.Substitute t: 2(h/160) / (1 - (h/160)^2) = h/60.Solve → h = 80 m. Verification / Alternative check: With h = 80, tan θ = 80/160 = 1/2. tan 2θ = 2*(1/2)/(1 - 1/4) = 1/(3/4) = 4/3. At 60 m, h/60 = 80/60 = 4/3 ✔️. Why Other Options Are Wrong: 100, 160, 200 m fail the tan 2θ equality when checked; only 80 m satisfies both positions. Common Pitfalls: Forgetting that observer distance changes to 60 m; sign errors in the tan 2θ formula; algebraic mistakes when clearing denominators. Final Answer: 80 m

Question 7

Depression changes from 30° to 60°: From the top of a 100 m tower, the angle of depression of a car is first 30° and later 60°. Assuming the car moves in a straight line toward the tower at constant level, find the distance the car travels between t…

Accepted Answer

Introduction / Context: Angles of depression equal angles of elevation from the same horizontal line. We compare two right triangles formed by the car’s positions and the tower height. Given Data / Assumptions: Tower height H = 100 m. First depression = 30°; second depression = 60°. Car moves horizontally toward the tower on level ground. Concept / Approach: Horizontal distance d = H / tan(depression). The travel distance equals d1 - d2 between the two instants. Step-by-Step Solution: d1 = 100 / tan 30° = 100 √3.d2 = 100 / tan 60° = 100 / √3.Travel = d1 - d2 = 100 √3 - 100/√3 = (100(3 - 1))/√3 = 200/√3 = 200 √3 / 3. Verification / Alternative check: Numeric: √3 ≈ 1.732 → 200/1.732 ≈ 115.47 m. This matches 200 √3 / 3 ≈ 115.47 m. Why Other Options Are Wrong: 100 √3 and 200 √3 are too large; 100 √3 / 3 is half the correct value; only 200 √3 / 3 matches the geometry. Common Pitfalls: Using cot instead of tan (or forgetting that depression equals elevation), subtracting in wrong order, or mixing degrees and radians in calculators. Final Answer: 200 √3 / 3

Question 8

Simplify the trigonometric expression (0° < A < 90°): Evaluate  sin A / (1 + cos A)  +  sin A / (1 - cos A)  and choose the correct simplified form.

Accepted Answer

Introduction / Context: This item asks for algebraic simplification of a standard trigonometric sum involving sin A and cos A. The interval ensures all expressions are defined (excluding division by zero cases). Given Data / Assumptions: Expression: sin A/(1 + cos A) + sin A/(1 - cos A). Angle A in (0°, 90°). Concept / Approach: Put both terms over a common denominator, use 1 - cos^2 A = sin^2 A, and simplify carefully. Step-by-Step Solution: Common denominator = (1 + cos A)(1 - cos A) = 1 - cos^2 A = sin^2 A.Numerator = sin A (1 - cos A) + sin A (1 + cos A) = sin A (2) = 2 sin A.Thus value = (2 sin A) / (sin^2 A) = 2 / sin A = 2 cosec A. Verification / Alternative check: Pick A = 30°: LHS = (0.5/1.866...) + (0.5/0.134..) ≈ 0.268 + 3.732 ≈ 4.0; RHS = 2 * cosec 30° = 2 * 2 = 4 ✔️. Why Other Options Are Wrong: 2 sec A, 2 sin A, 2 cos A mismatch after the Pythagorean substitution and do not hold numerically for test angles. Common Pitfalls: Forgetting the identity 1 - cos^2 A = sin^2 A; sign errors when expanding; canceling incorrectly before combining fractions. Final Answer: 2 cosec A

Question 9

Leaning ladder against a wall: A ladder makes a 60° angle with the horizontal ground, and its foot is 4.6 m away from the wall. Find the length of the ladder.

Accepted Answer

Introduction / Context: A ladder against a wall forms a right triangle with the ground (adjacent) and wall (opposite). The angle with ground and the horizontal distance to the wall are known; find the hypotenuse (ladder length). Given Data / Assumptions: Angle with ground = 60°. Adjacency (foot to wall) = 4.6 m. Ladder is straight; wall and ground are perpendicular. Concept / Approach: Use cos θ = adjacent / hypotenuse → hypotenuse = adjacent / cos θ. Step-by-Step Solution: cos 60° = 1/2.Length L = 4.6 / (1/2) = 9.2 m. Verification / Alternative check: Compute vertical reach: L * sin 60° = 9.2 * (√3/2) ≈ 7.97 m; Pythagoras with 4.6 m confirms hypotenuse ≈ 9.2 m. Why Other Options Are Wrong: 2.3 and 4.6 m are too short; 7.8 m corresponds to cos near 0.59, not 0.5. Common Pitfalls: Using tan instead of cos; mixing degrees with radians; rounding too early on √3 or cos values. Final Answer: 9.2 m

Question 10

Depression to tower top and bottom from a cliff: From the top of a cliff of height h metres, the angles of depression of the top and the bottom of a nearby vertical tower are 30° and 60° respectively. Find the height of the tower in terms of h.

Accepted Answer

Introduction / Context: Two angles of depression from a higher point to two points on or above the ground allow expressing horizontal distance and relative verticals, yielding the unknown tower height as a function of h. Given Data / Assumptions: Cliff height = h. Angles of depression: 30° to tower top, 60° to tower bottom. Tower is vertical; ground is level. Concept / Approach: If horizontal distance from cliff foot to tower base is x, then tan 60° = h / x → x = h/√3. For the top, vertical drop is h - T (T = tower height). tan 30° = (h - T)/x → solve for T. Step-by-Step Solution: x = h / √3 from tan 60°.tan 30° = (h - T)/x = 1/√3.Thus h - T = x/√3 = (h/√3)/√3 = h/3.So T = h - h/3 = 2h/3. Verification / Alternative check: Insert T back: Top drop = h - 2h/3 = h/3 → tan 30° = (h/3)/(h/√3) = 1/√3 ✔️. Why Other Options Are Wrong: h√3, 2h√3, h/3 mismatch the tangent relations; only 2h/3 satisfies both angles simultaneously. Common Pitfalls: Confusing elevation with depression; misplacing T in h − T; forgetting that both rays share the same horizontal distance x. Final Answer: 2h/3

Question 11

Complementary angles from two points on one line: From two points on the same straight line through the foot of a vertical tower, at distances a and b (a > b), the angles of elevation of the top are complementary. Find the height of the tower.

Accepted Answer

Introduction / Context: When two angles of elevation sum to 90°, one is θ and the other is 90° − θ. Using tan and cot relations gives a clean product form for the height. Given Data / Assumptions: Distances: a and b from the tower foot on the same line. Angles: θ at distance a, and (90° − θ) at distance b. Tower height = H. Concept / Approach: tan θ = H / a and tan(90° − θ) = cot θ = H / b. Multiply the equations and eliminate tan/cot to solve for H. Step-by-Step Solution: From a: H = a tan θ.From b: H = b cot θ = b / tan θ.Multiply: H^2 = (a tan θ)(b / tan θ) = ab → H = √(ab). Verification / Alternative check: Choose θ = 45°: then a = b = H, giving H = √(H^2) = H, consistent. For a ≠ b, the formula still satisfies both equations. Why Other Options Are Wrong: Forms with a^2 ± b^2 do not satisfy both tangent relations at once; √a(a − b) lacks symmetry expected from complementary angles. Common Pitfalls: Forgetting cot θ = 1 / tan θ; taking square root without noting H > 0; assuming a specific angle instead of using identities. Final Answer: √(ab)

Question 12

Two successive elevations after walking a: From a point A, the angle of elevation of the top of a vertical tower is α. After walking a metres straight toward the tower, the angle becomes β (β > α). Express the height of the tower in terms of a, α…

Accepted Answer

Introduction / Context: This is the standard two-elevation formula for a vertical tower when an observer moves a known distance a toward the tower, changing the angle from α to β. A sine-rule derivation yields a compact expression for the height. Given Data / Assumptions: Initial elevation = α; after walking a toward tower, elevation = β. Height of tower = H. Both observations on the same straight line toward the tower. Concept / Approach: Let initial horizontal distance be x. Then tan α = H / x and tan β = H / (x − a). Eliminating x gives H = a * tan α * tan β / (tan β − tan α). Using the sine addition identity, this equals a * sin α * sin β / sin(β − α). Step-by-Step Solution: tan α = H / x → x = H / tan α.tan β = H / (x − a) → x − a = H / tan β.Subtract: a = H(1/ tan α − 1/ tan β) = H( (tan β − tan α)/(tan α tan β) ).Thus H = a * tan α * tan β / (tan β − tan α) = a * sin α * sin β / sin(β − α). Verification / Alternative check: Plug small numeric values (e.g., α = 30°, β = 60°, a = known) to confirm both tan-form and sine-form give identical H. Why Other Options Are Wrong: They have wrong sign in denominator, extra factors, or inverted ratios; only option (a) matches the proven identity. Common Pitfalls: Using degrees with calculator in radian mode; mixing tan-based and sin-based expressions mid-derivation; forgetting that β > α so denominator is positive. Final Answer: a Sin α Sin β/ Sin(β - α)

Question 13

Changing depression from 30° to 45°: From the top of a vertical tower, a car approaches in a straight line on level ground. The angle of depression changes from 30° to 45° in 12 minutes. How much additional time is required (after it is 45°) for the…

Accepted Answer

Introduction / Context: With constant horizontal speed, times are proportional to horizontal distances. Angles of depression give those distances via tangent with the fixed tower height canceling out in ratios. Given Data / Assumptions: Depression changes 30° → 45° in 12 min. Motion along the line toward the tower; level ground; constant speed. Concept / Approach: Let tower height be H (cancels). Horizontal distances: d30 = H / tan 30° = H√3; d45 = H / tan 45° = H. The 12 minutes correspond to d30 − d45. From 45° to the tower is d45. Time scales with distance at constant speed. Step-by-Step Solution: Travel in 12 min: Δ = H√3 − H = H(√3 − 1).Speed v = Δ / 12 (arbitrary H cancels).Remaining distance from 45°: d45 = H.Time needed T = d45 / v = H / (H(√3 − 1)/12) = 12/(√3 − 1) = 6(√3 + 1).Numeric: √3 ≈ 1.732 → T ≈ 6 * 2.732 = 16.392 min ≈ 16 min 23 sec. Verification / Alternative check: Ratio method: T / 12 = d45 / (d30 − d45) = H / (H(√3 − 1)) → same value. Why Other Options Are Wrong: They correspond to rounding or formula mistakes; only ~16:23 matches exact evaluation of 6(√3 + 1) minutes. Common Pitfalls: Using depression distances incorrectly; forgetting to rationalize; rounding √3 too early leading to wrong seconds. Final Answer: 16 min 23 sec.

Question 14

Equal-height towers seen from a point between them: An observer stands between two vertical towers of equal height. The angles of elevation of the nearer and farther tops are 60° and 30° respectively. If the distance to the nearer tower is 100 m, fi…

Accepted Answer

Introduction / Context: A pair of equal-height towers on a straight line with the observer allows immediate height recovery from the nearer 60° sight, then the far distance from the 30° sight, giving the separation by addition. Given Data / Assumptions: Near top elevation = 60° at 100 m. Far top elevation = 30°; observer lies between towers. Towers have equal height H. Concept / Approach: Use tan for each: from near tower, H = 100 * tan 60°; from far tower, with distance D_far, H = D_far * tan 30°. Sum near and far horizontal distances to get total gap. Step-by-Step Solution: H = 100 * √3.For the far one: tan 30° = H / D_far → D_far = H / tan 30° = (100√3) * √3 = 300 m.Distance between towers = 100 + 300 = 400 m. Verification / Alternative check: Check near: H/100 = √3; far: H/300 = 1/√3 → both angles match 60° and 30° respectively. Why Other Options Are Wrong: Pairs with 100/√3 contradict tan 60°; a total of 300 m separation ignores the far geometry; only “100√3 m and 400 m” is consistent. Common Pitfalls: Assuming the observer is outside the segment; swapping near/far distances; mixing up tan 30° with cot 30°. Final Answer: 100√3 m and 400 m

Question 15

Man walking away from a lamp post: A lamp post is 5 m high. A 2 m tall man walks directly away from the post at 6 m/min. At what rate is the length of his shadow increasing?

Accepted Answer

Introduction / Context: Similar triangles relate the lamp, the man, and the tip of the shadow. Differentiating this relation with respect to time gives the rate of change of the shadow length. Given Data / Assumptions: Lamp height = 5 m; man height = 2 m. Man walks away from the lamp at dx/dt = 6 m/min. Let shadow length be s (from man to tip). Concept / Approach: The geometry yields a linear relation between s and the distance x of the man from the lamp. Then compute ds/dt via differentiation. Step-by-Step Solution: By similarity: 5 / (x + s) = 2 / s.Cross-multiply: 5s = 2x + 2s → 3s = 2x → s = (2/3) x.Differentiate: ds/dt = (2/3) dx/dt = (2/3) * 6 = 4 m/min. Verification / Alternative check: Pick x = 30 m → s = 20 m. After 1 min, x increases by 6 m → s increases by 4 m, matching the derivative. Why Other Options Are Wrong: 8 and 9 m/min contradict the linear 2/3 factor; 14 m/min is non-physical for these heights and speed. Common Pitfalls: Using 5/x = 2/s (omitting x + s); differentiating incorrectly; forgetting that s is measured from the man to the tip, not from the lamp. Final Answer: 4 m/min

Question 16

Shadow equal to height: When the length of the shadow of a vertical pole equals the height of the pole on level ground, what is the angle of elevation of the light source?

Accepted Answer

Introduction / Context: Relating a pole, its shadow, and the angle of elevation uses basic trigonometry: tan θ = opposite/adjacent = height/shadow length. Given Data / Assumptions: Height = shadow length. Level ground; vertical pole. Concept / Approach: tan θ = height/shadow = 1 → θ = 45°. Step-by-Step Solution: Let height = h and shadow = h.tan θ = h / h = 1 → θ = 45°. Verification / Alternative check: In a 45°-45°-90° triangle, legs are equal; consistent with the condition. Why Other Options Are Wrong: 30° or 60° would give unequal legs (shadows √3 h or h/√3), not equal lengths. Common Pitfalls: Using sine or cosine instead of tangent; misreading “shadow equals height”. Final Answer: 45°

Question 17

Broken pole touches ground at 30°: A vertical pole breaks at some point, and the top touches the ground at a point 21 m from the foot of the pole. The broken top makes an angle of 30° with the ground. Find the original total height of the pole.

Accepted Answer

Introduction / Context: After breaking, the upper segment forms the hypotenuse of a right triangle with the ground. The remaining standing segment equals the vertical component of that hypotenuse. The original height is the sum of the two segments. Given Data / Assumptions: Horizontal distance from foot to tip = 21 m. Inclination of top with ground = 30°. Let broken top length be L; remaining vertical be y. Concept / Approach: Resolve L into horizontal and vertical components: L cos 30° = 21 and L sin 30° = y. Total original height H = y + L (standing part + broken top length). Step-by-Step Solution: L cos 30° = 21 → L = 21 / cos 30° = 21 * 2 / √3 = 42/√3.y = L sin 30° = (42/√3) * (1/2) = 21/√3.H = y + L = (21/√3) + (42/√3) = 63/√3 = 21 √3. Verification / Alternative check: Numeric: √3 ≈ 1.732 → H ≈ 36.37 m; geometry checks out with the given angle and reach of 21 m. Why Other Options Are Wrong: 21 m is just the horizontal reach; 21/√3 is only the standing part; “None” is unnecessary since a clean expression exists. Common Pitfalls: Adding vertical components only; forgetting that the original top length equals the current slanted length; mixing degrees/radians. Final Answer: 21 √3 m

Question 18

Shadow difference for sun altitudes 30° and 60°: A vertical tower on level ground casts shadows when the sun’s altitude is 30° and 60°. The longer shadow is 50 m more than the shorter one. Find the height of the tower.

Accepted Answer

Introduction / Context: Shadow length s = H cot θ for a vertical object. Two altitudes give two shadows; their difference is known, allowing H to be solved exactly. Given Data / Assumptions: Altitudes: 30° and 60°. Difference in shadow lengths = 50 m. Tower height = H. Concept / Approach: s30 = H cot 30° = H √3; s60 = H cot 60° = H / √3. Their difference equals 50 → solve for H. Step-by-Step Solution: s30 − s60 = H(√3 − 1/√3) = 50.√3 − 1/√3 = (3 − 1)/√3 = 2/√3.Thus H * (2/√3) = 50 → H = 25 √3 m. Verification / Alternative check: Compute shadows with H = 25√3: s30 = 75; s60 = 25 → difference 50 ✔️. Why Other Options Are Wrong: 20√3 and 25/√3 give wrong differences; duplicate 20√3 is a distractor. Common Pitfalls: Using tan instead of cot for shadow; subtracting in reverse (negative); arithmetic slip when simplifying 2/√3. Final Answer: 25 √3 m

Question 19

Guy wire to a 20 m pole: A 20 m vertical electric pole is anchored to the ground by a wire attached to its top. If the wire makes a 60° angle with the horizontal ground, find the length of the wire.

Accepted Answer

Introduction / Context: The wire forms the hypotenuse of a right triangle with the ground (adjacent) and the pole (opposite). The angle given is with the horizontal, so use sine to relate opposite and hypotenuse. Given Data / Assumptions: Pole height = 20 m. Angle with ground = 60°. Concept / Approach: sin θ = opposite / hypotenuse → hypotenuse = opposite / sin θ. Step-by-Step Solution: Wire length L = 20 / sin 60° = 20 / (√3/2) = 40/√3 m. Verification / Alternative check: Numeric: √3 ≈ 1.732 → L ≈ 23.094 m; geometry consistent with a steep wire. Why Other Options Are Wrong: 40√3 is too long (would imply sin 60° < 0.5); 20√3 and 20/√3 arise from mixing cosine or tangent incorrectly. Common Pitfalls: Using cos instead of sin since angle is with ground; rationalizing incorrectly; forgetting the 1/2 factor in sin 60°. Final Answer: 40/ √3 m

Question 20

When shadow length equals height, find elevation: If the length of a pole’s shadow on level ground equals the pole’s height, what is the angle of elevation of the light source?

Accepted Answer

Introduction / Context: Same principle as similar items: tan θ = height/shadow. Equal height and shadow implies tan θ = 1 and θ = 45°. Given Data / Assumptions: Height = shadow; vertical pole; level ground. Concept / Approach: Use tan θ = opposite/adjacent. Step-by-Step Solution: tan θ = h / h = 1 → θ = 45°. Verification / Alternative check: A 45°-45°-90° triangle has equal legs, matching the condition. Why Other Options Are Wrong: 30° and 60° correspond to √3 ratios, not equality; 75° would make a very short shadow. Common Pitfalls: Using sine/cosine instead of tangent; misinterpreting which angle is asked (elevation, not depression). Final Answer: 45°

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