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Trigonometry is a branch of science or mathematics that deals with calculating the heights, distances, or angles of any triangle. It also helps understand the relationship between the heights and distance in a triangle. Trigonometry was first introduced in astronomy, and then it gained great importance in all other fields such as geography and navigation. In astronomy, trigonometry helps in the distance of Earth to the planets and stars. Moreover, heights and distances in trigonometry help us in the calculation of solving daily problems like finding the distance between two or more objects or the heights between mountains and hills.
Terms Related to Height and Distance in Trigonometry
Line of Sight
Line of sight is the imaginary line drawn from the eye of an observer towards the object. The line of sight gives us an idea about the position where the observer is viewing. For example, for a girl watching a cat, the imaginary dotted line from the eye of the girl to the cat will be the line of sight.
The Angle of Elevation
The elevation angle is observed when a person views an object upwards. It is the angle formed by the line of sight and the horizontal line only when the viewer or the observer watches upward—for example, a girl watching the cat sitting on the wall (i.e. on an elevation).
The Angle of Depression
The angle of depression is opposite of the angle of elevation, i.e. when a viewer or an observer watches downwards. The angle of depression is the angle between the line of sight and the horizontal line joining an observation point below the horizontal level—for example, a girl watching the cat on the street from the balcony.
We can measure the angle of elevation and depression by a special instrument based on the principles of trigonometry. This special instrument is the theodolite, which uses rotating telescopes to measure the angles.
Important Formulas for Trigonometry
There are six trigonometric ratios, i.e. sine, cosine, tangent, cotangent, secant, & cosecant.
- sin θ = Perpendicular / Hypotenuse
- cos θ = Base / Hypotenuse
- tan θ = Perpendicular / Base
- cosec θ = Hypotenuse/ Perpendicular
- sec θ = Hypotenuse/ Base
- cot θ= Base/ Perpendicular
Trigonometric Table
Angles | 0° | 30° | 45° | 60° | 90° |
sine | 0 | 1/2 | 1/√2 | √3/2 | 1 |
cosine | 1 | √3/2 | 1/√2 | 1/2 | 0 |
Tangent | 0 | 1/√3 | 1 | √3 | Not defined |
Cosecant | Not defined | 2 | √2 | 2/√3 | 1 |
Secant | 1 | 2/√3 | √2 | 2 | Not defined |
Cotangent | Not defined | √3 | 1 | 1/√3 | 0 |
Heights and Distances In Trigonometric Questions
Question 1- A girl standing near a wall observes the cat at an angle of elevation of 60∘. She walks 30 yards away from the wall, the angle of elevation becomes 30∘. Calculate the height of the wall.
Solution:
Let the height= h, and d be the distance between the girl and the wall.
tan60°=√3
h/d=√3
d=h/√3
tan30°=1/√3
h/d+30= 1/√3
√3h=d+30
√3h=h/√3+30
h(√3−1/√3)=30
h=15√3yd
Answer: ≈26yd
Question 2: There were two ships observed from a lighthouse, the two angles of depression are 30 and 45 degrees. Calculate the lighthouse’s height if these two ships are 100m apart.
Solution:
Applying the formula: Height = Distance / [cotangent (original angle) – cotangent (final angle)]
Height = 100 / (cot 30 – cot 45)
= 100 / (√3– 1)
=100(1.732-1)
=141.421m
Answer: The height of the lighthouse, if these two ships are 100m apart is =141.421m
Question 3- A ladder is of length 15 m. On slanting, it makes an angle of 60o with the wall. Calculate the height of the point where the ladder touches the wall using trigonometric formulas.
Solution:
cos θ = Base / Hypotenuse
cos θ = a/15
As we know that cos 60°= ½
½= a/15
On cross-multiplication, we get,
a= 15/2 m
a= 7.5m
Answer: The height of the point where the ladder touches the wall is 7.5m.
Conclusion
Trigonometry is the branch that deals with the determination of distances between two objects, heights or the angles between two points. Heights and distances in trigonometry help us calculate daily problems like finding the distance between two or more objects or the heights between mountains and hills. The two important terms related to trigonometry are the angle of elevation and angle of depressions. The angle of elevation is observed when a person views an object upwards, whereas the angle of depression is formed when a viewer or an observer watches downwards. There are six trigonometric ratios, i.e. sine, cosine, tangent, cotangent, secant, & cosecant. The formulae of these ratios are given above.
