TRIGONOMETRY

TRIGONOMETRY “The deep study of nature is the most fruitful source of mathematical discoveries” - Joseph Fourier. Francois Viete ( – AD(CE)) Indian scholars of the 5th century AD(CE), realized that working with half-chords for half-angles greatly simplified the theory of chords and its application to astronomy. Mathematicians like Aryabhata, the two Bhaskaras and several others developed astonishingly sophisticated techniques for calculating half-chord (Jya) values. Mathematician Abu Al-Wafa of Baghdad believed to have invented the tangent function, which he called the “Shadow”. Arabic scholars did not know how to translate the word Jya, into their texts and simply wrote jiba as a close approximate word. Misinterpreting the Arabic word ‘jiba’ for ‘cove’ or ‘bay’, translators wrote the Arabic word ‘jiba’ as ‘sinus’ in Latin to represent the half-chord. From this, we have the name ‘sine’ used to this day. The word “Trigonometry” itself was invented by German mathematician Bartholomaeus Pitiscus in the beginning of 17th century AD(CE). Recall Trigonometric Ratios Let ° < < ° Hypotenuse Adjacent Side P M O Opposite side Fig. . Let us take right triangle OMP sin q = Opposite side Hypotenuse MP OP cos q = Adjacent side Hypotenuse OM OP From the above two ratios we can obtain other four trigonometric ratios as follows. tan ; cot ; cosec q ; sec q Trigonometric ratios of complementary angle sin( ° − cos( ° − tan( cot ° − cosec( ° − sec sec( cosec ° − cot( tan ° − Visual proof of trigonometric complementary angle Consider a semicircle of radius as shown in the figure. Let ∠ QOP q . Then ∠ ° − QOR q , so that OPQR forms a rectangle. From triangle OPQ , OP OQ = cos q