Some Applications of Trigonometry · Class 10 Mathematics: quick revision, 35 bites

Why this works

Measure an angle, not the tower

One side and one acute angle fix a right triangle completely.

Takeaway. Angles reach where tape measures cannot.

Line of sight

The line you look along

The straight line from the eye to the object.

Takeaway. Sight line means hypotenuse.

Angle of elevation

Eyes up

The angle from the HORIZONTAL up to the line of sight, when the object is above you.

Takeaway. Always measured from the horizontal.

Angle of depression

Eyes down

The angle from the HORIZONTAL down to the line of sight, when the object is below you.

Takeaway. Still measured from the horizontal, not the vertical.

The horizontal you must add

Draw the dashed line

Looking down from a cliff, there is no ground at eye level for the angle to rest on.

Takeaway. No horizontal, no angle of depression.

Depression equals elevation

The same angle both ways

The horizontal at the top and the horizontal at the bottom are parallel.

Takeaway. Move the angle to whichever end makes the triangle easier.

Tangent links the two legs

The workhorse

Takeaway. If the sight line is not mentioned, it is a tangent question.

Pick by what you have

Which ratio?

Takeaway. Name the three sides first; the ratio then picks itself.

45° means equal

The easy angle

At 45° the height equals the horizontal distance, always.

Takeaway. Equal legs at 45 degrees.

The 30° sight line

Twice the height

At 30° the line of sight is exactly twice the height, since $\sin 30° = \frac12$.

Takeaway. A useful shortcut when the hypotenuse is wanted.

30 m away, 30° up

A first height

Takeaway. About 17.3 m.

20 m high, 45° up

A first distance

Takeaway. Equal, as 45 degrees always gives.

A kite at 60°

How long is the string?

The string is the line of sight, so use sine, not tangent.

Takeaway. Hypotenuse wanted means sin or cos.

Rationalise the denominator

Finish the surd

$\frac{30}{\sqrt3}$ is not a finished answer.

Takeaway. No surd is left downstairs.

Adding the observer

Height above what?

If the angle was taken at eye level, the triangle gives the height ABOVE THE EYE.

Takeaway. Ask what the triangle measured from.

Laying out the answer

Where the marks are

Takeaway. The diagram carries marks of its own.

What the syllabus allows

Only three angles

Every angle in this chapter is 30°, 45° or 60°, and a problem uses at most two right triangles.

Takeaway. So the table from Chapter 8 is all the arithmetic you need.

Everest, 1852

A mountain measured from afar

Everest's height was first computed from angle readings taken more than 100 miles away.

Takeaway. The same triangle, on a very large scale.

Why two triangles

One look is not enough

One observation gives one equation but leaves two unknowns, the height and the distance.

Takeaway. Two unknowns need two angles.

The shared vertical

The side they have in common

Both right triangles stand on the same vertical line — the height itself.

Takeaway. Eliminate the height first.

Larger angle, nearer point

Closer means steeper

The same height seen from closer up subtends a larger angle.

Takeaway. Use it to set up, and again to check.

Distances differ by the gap

Same side

If both observers are on the same side of the tower, one is $d$ away and the other $d + \text{gap}$.

Takeaway. Subtraction, not addition.

Distances add to the gap

Opposite sides

If the tower stands BETWEEN the two points, the distances are $x$ and $\text{gap} - x$.

Takeaway. Look for the word 'between'.

Flagstaff on a tower

One point, two objects

Both triangles share the same horizontal distance and the same observer.

Takeaway. Subtract, do not stop at the taller triangle.

60° and 30°, 20 m apart

The classic

Takeaway. Height 10√3 ≈ 17.3 m.

80 m of road

Between two points

Takeaway. Height 20√3, distances 20 m and 60 m.

45° then 60°, 20 m away

How tall is the flag?

Takeaway. About 14.6 m.

The flagstaff trap

Do not stop early

$20\sqrt3$ is the height of the tower AND flagstaff together.

Takeaway. Read what was actually asked for.

Use the conjugate

Clearing a stubborn surd

The 30°–45° pairing leaves $\frac{40}{\sqrt3 - 1}$.

Takeaway. Multiply by the conjugate over itself.

30° becomes 45°

Walking towards it

A moving observer is just two points on the same side.

Takeaway. Same picture, different story.

Keep the surds back

Substitute late

Arrange the algebra first and put the standard values in afterwards.

Takeaway. Algebra first, arithmetic second.

Draw them together

One figure, not two

Draw the shared vertical once and both lines of sight to it.

Takeaway. The single figure is what shows the gap.

Why the angles must differ

Equal angles give nothing

If both angles were the same, the two equations would be identical and eliminating $h$ would leave $0 = 0$.

Takeaway. Different angles are what make it solvable.

Two quick tests

Check before you stop

Takeaway. Both take five seconds.

Triangulation

A country mapped by triangles

Nineteenth-century surveys chained triangles across India, each new height fixed from two known points.

Takeaway. The same two equations, millions of times.