# S.L. Loney Height and Distances Solutions (Chapter 14)

SL Loney Plane Trigonometry Solutions for Chapter 14 ‘Heights and Distances’ is an extension of Chapter of SL Loney Solutions book which had simpler problems on Heights and Distances. With the help of these solutions, you will learn how to find the horizontal distance of an object from the point of ascent, find the height of an object and distance between two objects. All these questions are of a much-advanced level. They will be immensely helpful in appearing for prestigious entrance exams like IIT-JEE and NEET and also give you a better chance of pursuing a career in advanced mathematics.

Compete Solutions for SL Loney Plane Trigonometry for ‘Heights and Distances’ has a total of 63 questions and 2 exercises. These problems are mostly based on the concept of land-surveying. Methods and formulas are given to find the height of an inaccessible tower by observing it from different distant points and using angles of elevation. There is also a discussion on bearings and points of a compass.

You will get detailed Solutions to SL Loney Plane Trigonometry by our academic experts at Instasolv on our easily accessible online portal. The solutions keep in mind the level of difficulty of a problem and are handled in a way that makes it look less daunting and more manageable. Now you can easily prepare for tough competitive exams like JEE.

## Important Topics for SL Loney Plane Trigonometry Solutions Chapter 14: Heights and Distances

The traditional method of surveying land and measuring positions on the earth’s surface uses the angles and distances between locations with instruments like theodolites and tapes. There are also instruments like “total stations” which combine the distance measurement and angle measurements into one unit to determine positions.

**Finding the Height of an Inaccessible Tower – **This can be done by observing the tower from different points on the ground and making use of angles of elevation and distance covered, in calculating its height. Let us consider the below image:

If point P is where the height of the tower is with an unknown height of x, we draw a line horizontal from the base of the tower Q to point A and measure the angle of elevation at that point (say α), then a distance AB (say a) is travelled directly towards the foot of the tower Q and another angle of elevation is taken at B (Say β). Now using the trigonometric functions for triangle PBQ we get:

Sin β = x/BP –> (i)

And from triangle PAB we get:

Sin PAB/Sin PBA = PB/ α = Sin α/Sin (β – α) -> (ii)

We apply multiplication to equations (i) and (ii) to get:

x/ α = (Sin α Sin β)/ Sin (β – α)

So x = α * (Sin α Sin β)/ Sin (β – α)

This way we get the value of x which is fit for logarithmic calculations.

**Finding Distances between 2 Inaccessible Points – **This can be done by making observations at 2 points whose distance is known and then joining them. All 4 points must lie in the same plane.

In the above figure if P and Q are not accessible, in order to find PQ we take 2 points A, B the distance between them is known to be a. We take the angles of the inaccessible point at A as PAB and QAB; α and β respectively. Then we measure angles from point Q, PBA, and QBA and let them be and ϓ respectively. We draw the following relation from triangle PAB:

AP/ α = Sin ϓ/Sin APB = Sin ϓ/Sin(ϓ + α)

And from triangle QAB we get:

AQ/ α = Sin β /Sin( + β)

Now we know sides AP and AQ in the triangle APQ and the included angle PAQ is (α – β) hence we can apply trigonometric formulas to get PQ.

If the points were not in the same plane then angle PAQ will not be (α – β) so we need to additionally measure that angle to find the height of PQ.

**Compass Bearings and Points – **A compass is based on a circle and used to find the direction of a point. 4 cardinal points cut the circle into 4 halves giving North, East, South, and West (NESW) and 4 primary internal cardinal points cut the circle in further 8 halves giving North East, South East, South West, and North West (NE, SE, SW, NW). Then they are supplemented by 8 Secondary intercardinal points which divide the circle into 16 halves given by North northeast, East northeast, East south-east, South south-east, South south-west, West south-west, West north-west, and North north-west. These points are used by marines to find out safe passages.

- North represents 0 or 360 degrees
- East denotes 90 degrees
- South is at 180 degrees
- West represents 270 degrees
- The bearing of a point is measured by the angle in degrees which is in a clockwise direction from the north line.
- Bearing tells us the direction of one point with respect to another point

**Example of Measuring Bearing – **In the figure below we can get the bearing of point A relative to the point B by measuring the angle when A is seen from B, which is shown as 065 degrees.

Bearing is also mentioned in terms of direction, so the line BA bisects the angle between north and east hence its bearing is North-east. All bearings are measured in a horizontal plane.

### Discussion of Exercises of SL Loney Plane Trigonometry Solutions Chapter 14: Heights and Distances

- The first part of the exercise has 22 questions dealing with measurements of heights and distance such as a flagstaff (by travelling a certain distance in its direction), a tower, a pole, etc. You will be using the angle of elevation along with trigonometric equations to solve them.
- The second part of the exercises has 40 questions where finding distance between a boat and a bridge gave the angle the boat subtends at the bridge, the altitude of sun by the length of the shadow it casts, few questions on bearing where a ship is sailing and one has to find its speed, and some problems where formulas of logarithm are also to be used. They are all word problems.

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