# S.L. Loney Simple Problem in Height And Distance Solutions (Chapter 3)

SL Loney Plane Trigonometry Solutions for Chapter 3 ‘Simple Problems in Height and Distances’ is a classic textbook for getting comprehensive subjective knowledge on objects of trigonometry. If you want to pursue an advanced mathematics course or appear in competitive exams like IIT JEE and NEET, you must have this book as it has all the solutions for SL Loney Solutions Simple Problems in Heights and Distances exercises. You will be able to solve questions related to finding the angular elevation, heights of objects when trigonometric ratios are given and angular depression.

SL Loney Plane Trigonometry Solutions for JEE ‘Simple Problems in Height and Distances’ has only 1 exercise with 21 questions which** **are based on finding the distances between points or object heights act without actually measuring the distances or heights. These questions are solved by applying trigonometric concepts like the angle of elevation, angle of depression to solve such sums. You will learn about what are theodolites and sextants and how to use them.

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## Important Topics for SL Loney Plane Trigonometry Solutions Chapter 3: Simple Problems in Height and Distances

Trigonometry is used in many important fields from engineering and architecture to astronomy. By assuming lines connecting points, one can apply trigonometric ratios to measure heights and distances.

**Height, distance, and triangle: **Measuring an object in vertical direction gives its height, and measuring an object from a specific point in horizontal direction gives its distance from that point. If an imaginary line is drawn from the observation point to the height or topmost point of the object being observed then height, distance, and this imaginary line will form a triangle.

**The Angle of Elevation – **This angle is relevant when the point of observation is at a greater height than the observer. In the above figure, AB is the height of the object, BC is the distance of observation point which is at C and AC is the line of sight. ∠AOC is the angle of elevation here. It is called an angle of elevation since the observer needs to raise (or elevate) his eyes from the horizontal level to the line of sight to look at the object.

**The Angle of Depression – **This angle applies when the point of observation is at a lower height than the observer. In the above figure if we consider the observer to be at point A and object is at point C then if a horizontal line is drawn from the observer’s eye A to the object C (let us say AD) then the angle ∠DAP is the angle of depression. It is called an angle of depression since the observer needs to lower (or depress) his eyes from the horizontal level to the line of sight to look at the object.

**Measuring Heights and Distances –** The directions of the objects are given by the angle of elevation and angle of depression. Trigonometric ratios (Sin and Cos are used to measure heights and distances).

- An instrument called Theodolite is used to measure the angle of elevation and depression. This instrument is based on the principles used in trigonometry. A simple form of it has a telescope attached to a piece of wood that can be rotated to measure horizontal and vertical angles.
- A sextant is used majorly as a navigating instrument on ships and can measure angular distances between objects.

**Hypotenuse – **The side opposite to 90 degrees in a right-angles triangle is called the hypotenuse.

**Heights and Distances Formulas**

**In any right-angled triangle:**

- Sin θ = opposite/hypotenuse
- Cos θ = adjacent/hypotenuse
- Tanθ = opposite/adjacent = Sin(θ)/Cos(θ)
- Sec θ = hypotenuse/adjacent = 1/cos θ
- Cosec θ = hypotenuse/opposite = 1/Sin θ
- Cot θ = adjacent/opposite = 1/tan θ

Here θ is the reference angle (a reference angle is any of the other 2 angles which are not 90 degrees). “Opposite” refers to the side of the triangle opposite to this reference angle and “adjacent” refers to the side of the triangle which is adjacent to the reference angle.

### Discussion of Exercises of SL Loney Plane Trigonometry Solutions Chapter 3: Simple Problems in Height and Distances

- The exercise for Heights and Distances t has 21 questions where you will be using various trigonometric ratios and formulas to find heights and distances.
- In some questions, you have to find the height of the tower or a tree while you are given the angle of elevation
- A few questions ask you to find the height of the objects and the point when the distance between 2 objects is given along with the angels they subtend at a point on the ground.
- In one question, you have to find the angle of elevation of the sun given length of the shadow of a pole.

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