3 Engineering Example 2
3.1 Horizon distance
Problem in words
Looking from a height of 2 m above sea level, how far away is the horizon? State any assumptions made.
Mathematical statement of the problem
Assume that the Earth is a sphere. Find the length of the tangent to the Earth’s sphere from the observation point .
Figure 8 :
Mathematical analysis
Using Pythagoras’ theorem in the triangle shown in Figure 8,
Hence
If m, then the variation of with is shown in Figure 9.
Figure 9
At an observation height of 2 m, the formula predicts that the horizon is just over 5 km away. In fact the variation of optical refractive index with height in the atmosphere means that the horizon is approximately 9% greater than this.
Task!
Using the triangle in Figure 10 which can be regarded as one half of the equilateral triangle , calculate sin, cos, tan for the angles and .
Figure 10
By Pythagoras’ theorem:
Hence
Values of and can of course be obtained by calculator. When entering the angle in degrees ( e.g. ) the calculator must be in degree mode. (Typically this is ensured by pressing the DRG button until ‘DEG’ is shown on the display). The keystrokes for are usually simply SIN 30 or, on some calculators, 30 SIN perhaps followed by =.
Task!
(a) Use your calculator to check the values of and obtained
in the previous Task.
(b) Also obtain . (’ denotes a minute )
(a) to 4 d.p.
(b) to 4 d.p., to 4 d.p.
3.2 Inverse trigonometric functions (a first look)
Consider, by way of example, a right-angled triangle with sides 3, 4 and 5, see Figure 11.
Figure 11
Suppose we wish to find the angles at and . Clearly so we need to solve one of the above three equations to find .
Using we write (read as ‘ is the inverse sine of ’)
The value of can be obtained by calculator using the ‘ ’ button (often a second function to the function and accessed using a SHIFT or INV or SECOND FUNCTION key).
Thus to obtain we might use the following keystrokes:
INV SIN 0.6 =
or
3 5 INV SIN =
We find (to 4 significant figures).
Key Point 5
Inverse Trigonometric Functions
(The alternative notations arc arc arc are sometimes used for these inverse functions.)
Task!
Check the values of the angles at and in Figure 11 above using the functions on your calculator. Give your answers in degrees to 2 d.p.
Task!
Check the values of the angles at and in Figure 11 above using the functions on your calculator. Give your answers in degrees to 2 d.p.
You should note carefully that does not mean .
Indeed the function has a special name – the cosecant of , written . So
(the cosecant function).
Similarly
(the secant function)
(the cotangent function).
Task!
Use your calculator to obtain to 3 d.p. (Use the sin, cos or tan buttons unless your calculator has specific buttons.)