Some Applications of Trigonometry · Class 10 Mathematics: 46 flashcards
All 46 flashcards from this kit, free and with no sign-up.
1. What is the line of sight?
Show the answer
The straight line from the observer's eye to the object; it is the hypotenuse of the right triangle.
2. Define the angle of elevation.
Show the answer
The angle between the horizontal through the eye and the line of sight, when the object is above the observer.
3. Define the angle of depression.
Show the answer
The angle between the horizontal through the eye and the line of sight, when the object is below the observer.
4. From what line are both angles measured?
Show the answer
The horizontal through the observer's eye — never the vertical.
5. Why does the angle of depression equal the angle of elevation?
Show the answer
The two horizontals are parallel and the line of sight is a transversal, so the angles are alternate interior angles.
6. Which ratio links the height and the ground distance?
Show the answer
Tangent: $\tan\theta = \frac{\text{height}}{\text{distance}}$.
7. Which ratio involves the line of sight and the height?
Show the answer
Sine, because the line of sight is the hypotenuse.
8. Which ratio involves the line of sight and the ground distance?
Show the answer
Cosine.
9. What is special about an elevation of 45°?
Show the answer
The height equals the horizontal distance.
10. At 30°, how does the line of sight compare with the height?
Show the answer
It is exactly twice the height, since $\sin 30° = \frac12$.
11. A tower is 30 m from an observer, elevation 30°. Find its height.
Show the answer
$h = 30\tan 30° = \frac{30}{\sqrt3} = 10\sqrt3 \approx 17.3$ m.
12. A tower is 20 m tall. At what distance is its elevation 45°?
Show the answer
20 m — at 45° the two legs are equal.
13. A kite is 60 m high and its string makes 60° with the ground. How long is the string?
Show the answer
$\ell = \frac{60}{\sin 60°} = \frac{120}{\sqrt3} = 40\sqrt3$ m.
14. Why must you draw a dashed horizontal for an angle of depression?
Show the answer
Because there is no ground at eye level for the angle to be measured from; without it the angle is taken from the vertical by mistake.
15. An angle is taken at an eye height of 1.5 m. What must you do at the end?
Show the answer
Add 1.5 m, because the triangle only gives the height above eye level.
16. Is $\frac{40}{\sqrt3}$ a finished answer?
Show the answer
No. Rationalise it to $\frac{40\sqrt3}{3}$.
17. Which angles does this chapter allow?
Show the answer
30°, 45° and 60° only.
18. How many right triangles may one problem use?
Show the answer
At most two.
19. A 40 m lighthouse has a boat at a depression of 45°. How far out is the boat?
Show the answer
40 m, since the elevation at the boat is also 45° and $\tan 45° = 1$.
20. What is the first thing to write in a heights-and-distances answer?
Show the answer
A labelled diagram.
21. If the sight line is not mentioned, which ratio is almost always needed?
Show the answer
Tangent.
22. A ladder at 60° to the ground reaches 6 m up a wall. How long is the ladder?
Show the answer
$\frac{6}{\sin 60°} = \frac{12}{\sqrt3} = 4\sqrt3$ m.
23. Height 15 m, elevation 60°. Find the ground distance.
Show the answer
$\frac{15}{\sqrt3} = 5\sqrt3$ m.
24. Does the angle of elevation halve when you double your distance?
Show the answer
No — angles do not scale like lengths.
25. Why can a single angle of elevation not give the height of a tower?
Show the answer
It gives one equation with two unknowns — the height and the distance.
26. What do the two right triangles in these problems always share?
Show the answer
The vertical side, which is the height.
27. Which point gets the larger angle of elevation?
Show the answer
The nearer one.
28. Two observers on the same side, gap $g$. How are the distances related?
Show the answer
They are $d$ and $d + g$ — they differ by the gap.
29. A tower between two points $g$ apart. How are the distances related?
Show the answer
They are $x$ and $g - x$ — they add to the gap.
30. How do you find the height of a flagstaff on a tower?
Show the answer
Find the tower-plus-flagstaff height and the tower height from the same point, then subtract.
31. Angles 60° and 30° from two points 20 m apart on the same side. Find the height.
Show the answer
$d\sqrt3 = \frac{d+20}{\sqrt3}$ gives $d = 10$, so $h = 10\sqrt3 \approx 17.3$ m.
32. A tower between two points 80 m apart, angles 60° and 30°. Find the height.
Show the answer
$x\sqrt3 = \frac{80-x}{\sqrt3}$ gives $x = 20$, so $h = 20\sqrt3 \approx 34.6$ m.
33. From 20 m away, the tower is at 45° and the flagstaff top at 60°. How tall is the flagstaff?
Show the answer
$20\sqrt3 - 20 = 20(\sqrt3 - 1) \approx 14.6$ m.
34. A man walks 40 m towards a tower and the elevation rises from 30° to 45°. Find the height.
Show the answer
$h = \frac{40}{\sqrt3 - 1} = 20(\sqrt3 + 1) \approx 54.6$ m.
35. How do you rationalise $\frac{40}{\sqrt3 - 1}$?
Show the answer
Multiply above and below by the conjugate $\sqrt3 + 1$, giving $\frac{40(\sqrt3+1)}{2} = 20(\sqrt3+1)$.
36. What happens if both given angles are equal?
Show the answer
The two equations become identical, so nothing can be found — the problem is unsolvable.
37. Should you substitute the standard values before or after eliminating the height?
Show the answer
After. Arranging the algebra first keeps the working clean.
38. Why draw one figure rather than two?
Show the answer
Because a single figure shows how the two distances are related through the gap.
39. What check confirms you assigned the angles correctly?
Show the answer
The larger angle must have come out with the smaller distance.
40. In a flagstaff problem, what do the two triangles share?
Show the answer
The same observer and the same horizontal distance.
41. How many right triangles may a syllabus problem use?
Show the answer
At most two.
42. Which angles are permitted?
Show the answer
30°, 45° and 60° only.
43. Angles 45° and 30° from two points 40 m apart on the same side. Find the height.
Show the answer
$d = \frac{40}{\sqrt3-1} = 20(\sqrt3+1)$, and $h = d$ since $\tan 45° = 1$, so $h = 20\sqrt3 + 20 \approx 54.6$ m.
44. Two observers on opposite sides of a 60 m road see a tower between them at 45° each. Where is the tower?
Show the answer
Exactly halfway, 30 m from each, and the height is 30 m.
45. What is the last thing to do in one of these solutions?
Show the answer
Rationalise, state the unit, and check the nearer distance is the smaller one.
46. A moving observer problem is really which type?
Show the answer
Two points on the same side — the two positions are the two points.
Your notes can be a kit like these
Every kit here was made from real study material. Point Veda at a chapter, a PDF, a photo or a link and get notes, flashcards, practice MCQs, games and a scored test — free, in 42 languages.
- Chapter notes
- Flashcards
- Practice MCQs
- Scored tests
- Exam prep
Get Veda free See what students get
Free to start · No ads · Works in 42 languages