In the s, when India was still a British colony, a British surveyor stationed not far away from the foothills of the Himalayas sighted an extremely high peak in the distance. When a mathematician calculated its height based on measurements taken from a survey, he realised that it was the world's highest mountain. It was then named Mount Everest, in honour of Sir George Everest, the Surveyor-General of India.
How was the mountain's height measured?
Firstly, a point on the ground away from the mountain was chosen.
Next, the distance from to the base of the mountain was measured as precisely as possible (this was extremely difficult to do due to hilly terrain).
The angle of elevation of the summit from the point was then measured.
Using trigonometry, the mountain's height was finally determined.
What, then, is trigonometry? Basically, it is the study of the relationship between angles and lengths.
a. Defining the trigonometric ratios at acute angles
Let be an acute angle, i.e. . Draw a right-angled triangle with and .
The sine, cosine and tangent ratios at the angle are then defined as follows:
To remember these definitions more easily, memorise the mnemonic 'toa-cah-soh' or 'soh-cah-toa'.
. Use the concept of similar triangles to show that the definitions of the three basic trigonometric ratios at an acute angle do not depend on the size of the right-angled triangle drawn.
. Show that, for acute , .
. Construct such that . Construct a second such that and . Use the triangles to determine the following values:
. Construct such that and . If , find the values of and .
. Show that, for acute angle , .
In the next note, Introduction to Trigonometry (coming soon!), we will define the three basic trigonometric ratios at other angles.