Trigonometry is fun!

We learn to do all kinds of difficult and complicated problems in school and to do them fast, but do we understand what we are doing?

We hardly know even very simple and handy calculations. The fact is trigonometry is a versatile and indispensable tool in science and in everyday life. People who did mathematics at school even long ago should be able to solve the given problems. In fact I would say anyone wanting to would be able to understand the solutions.

We learn what we learn to do by doing. Mathematics is learnt to do by doing it. Mathematics is in the first place an activity and cannot be learnt passively such as staring at a blackboard or TV screen or a tablet for that matter.

Note that one should make your own sketch for each of these problems and try first before looking at the solutions. In all these problems the distance cannot be directly measured. You see trigonometry can be very handy! And fun!

Sketches and Solutions are available on the Instagram account “full.force.gale”.

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page 1

Trigonometry, some very basics

Given a ninety-degree triangle, a given angle determines the adjacent and opposite sides, and the long side is the hypotenuse.

sin (angle) = opposite / hypotenuse ;

cos (angle) = adjacent / hypotenuse ;

tan (angle) = opposite / adjacent

For a given angle in degrees, the values for sin, cos, tan will be given by your scientific calculator or log tables if it must. In practice such an angle could be measured with some improvised protractor (e.g. a theodolite) and distances with in effect a ruler or such measuring stick or a line.

The idea is for the reader to first try the problems without looking at the solutions.

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Applications

page 2

A tree throws a shadow of 15meters and  the shadow's angle is 60deg (degrees) to the horizontal. How tall is the tree?

The method may be refined by planting an upright stick and measuring it's shadow and height, the two shadows' angles would be the same. So you can easily find the height of a given tree.

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page 3

Two trees along a river are 60m apart. Along their distance a rock on the other side measures angles both of 75deg. How wide is the river?

Answer: 111.96m

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page 4

A boy's kite is in the air on a straight string of fifty meters. The string makes an angle of 25deg to the horizontal. How high is the kite?

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page 5

For a right-angled triangle the theorem of Pythagoras gives,

adjacent^2 + opposite^2 = hypotenuse^2, where for a given number y is the square y^2 = y x y, and then the square root of y^2 is  y .                                              

As an application a rectangular fence is 40m by 30m , and the camp is divided in the diagonal giving two triangles. How long is the diagonal fence? You may use the Pythagoras theorem it gives 50meters.

You might also find the area of the fenced-off triangle.