PLANE TRIGONOMETRY

PLANE TRIGONOMETRY is based on the fact of similar figures. 

Figures are similar if they are equiangular and the sides that make the equal angles are proportional.

For triangles to be similar, however, it is sufficient that they be equiangular. From that it follows:

Right triangles will be similar if an acute angle of one is equal to an acute angle of the other.

In the right triangles ABC, DEF, if the acute angle at B is equal to the acute angle at E, then those triangles will be similar.  Therefore the sides that make the equal angles will be proportional.  If CA is half of AB, for example, then FD will also be half of DE.

Trigonometry and Right TrianglesFirst of all, think of a trigonometry function as you would any general function. That is, a

value goes in and a value comes out.The names of the three primary trigonometry functions are:

sinecosinetangent

These are abbreviated this way:sine.....sincosine.....costangent.....tan

So, instead of writing f(x) , we will write:sin(x)cos(x)tan(x)

Now, as usual, the input value is x. This input value usually represents an angle. For the sine function, when the input value is 30 degrees, the output value is 0.5. We would write that statement this way:

0.5 = sin(30 °)

The input value for these trigonometric functions is an angle. That angle could be measured in degrees or radians. Here we will consider only input angles measured in degrees from 0 degrees to 90 degrees. The output value for these trigonometric functions is a pure number. That is, it has no unit.

Let’s see the angle A. Notice that the sides of the triangle are labeled appropriately "opposite side" and "adjacent side" relative to angle A. The hypotenuse is not considered opposite or adjacent to the angle A.The value for the tangent of angle A is defined as the value that you get when you divide the opposite side by the adjacent side. This can be written:

tan (A) = opposite / adjacent

Suppose we measure the lengths of the sides of this triangle. Here are some realistic values:

This would mean that:tan(A) = opposite / adjacent = 4.00 cm / 6.00 cm = 0.6667

Indirect measurement.   

Trigonometry is used typically to measure things that we cannot measure directly.

For example, to measure the height h of a flagpole, we could measure a distance of 100 feet from its base.  From that point P, we could then measure the angle required to sight the top.  If that angle (called the angle of elevation) turned out to be 37°, then

  tan 37°=     h     100

 100 × tan 37°= hh =75.4 feet

Exercises:

1. From a boat we can see the light of a lighthouse with an angle of 24º. How far are we from the shore if the lighthouse is 80 metres high?

2. Find the angle of the sun rays with the horizontal line so that a streetlight 10,7 metres high casts a shadow of 3,6 metres.

3. At one particular moment, a tree casts a shadow 4,23 metres long. At the same time, the shadow of a stick (1,20 metres high) measures 0,64 meters. Find the height of the tree.