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TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES By M. Jaya krishna Reddy Mentor in mathematics, APIIIT-Basar, Adilabad(dt),A.P. India. Acute Angle: ACUTE ANGLE An angle whose measure is greater than zero but less than 90 is called an “acute angle” T E R M I N A L R A Y o Initial ray sin opp b hyp cos tan adj cos ec opp c a hyp sec hyp adj a opp b adj a adj a opp c c c cot A b hyp c b b Ѳ C a Commonly used mnemonic for these ratios : Some Old Houses Can’t Always Hide Their Old Age B History: •Trigonometric functions(also called circular functions) are functions of an angle. •They are used to relate the angles of a triangle to the lengths of the sides of a triangle. •Sumerian astronomers introduced angle measure, using a division of circles into 360 degrees. •The sine function was first defined in the “surya siddhanta” and its properties were further documented by the fifth century Indian mathematician and astronomer “Aryabhatta”. •By 10th century the six trigonometric functions were used. Applications: •In 240 B.C. a mathematician named “Eratosthenes” discovered the radius of the earth as 4212.48 miles using trigonometric functions.. •In 2001 a group of European astronomers did an experiment by using trigonometric functions and they got all the measurement, they calculate the Venus was about 105,000,000 km away from the sun and the earth was about 150, 000, 000 km away. •Optics and statics are 2 early fields of Physics that use trigonometry. •It is also the foundation of the practical art of surveying 1. Prove that cos . tan sin Sol: c o s . ta n c o s . 2. Prove that cos s in s in c o s s in . c o s s e c c o s e c s in c o s Sol: s in s e c c o s e c s in c o s 1 cos 1 s in s in c o s s in c o s s in . c o s sin . co s Fundamental Relations: opp sin hyp A b c c b co s adj h yp a c Ѳ C Squaring and adding both the equations (s in ) 2 (c o s ) 2 b c 2 a c 2 a 2 b2 a2 c2 2 1 2 c c s in B cos 2 1 From the above diagram, By Pythagorean Rule, a 2 b 2 c 2 A tan opp sec b adj a h yp c adj b a Squaring and subtracting the equations, we get (s e c ) 2 (ta n ) c c a 2 2 b a C 2 c2 b2 a2 2 1 2 a a sec 2 Ѳ ta n 2 1 a From the above diagram, By Pythagorean Rule, a Similarly, co sec cot 1 2 2 B 2 b 2 c 2 cos 2 1 2 2 s e c ta n 1 2 2 co sec cot 1 s in 2 sec 1 2 Example: Prove that sec sin sec 1 2 sol: G iven that sec tan 2 sec sin cos tan sec cos 1 sin Ex: Prove that sec2Ѳ - cosec2Ѳ = tan2Ѳ - cot2Ѳ Sol: We know that sec2Ѳ - tan2Ѳ = 1 = cosec2Ѳ - cot2Ѳ sec2Ѳ - tan2Ѳ = cosec2Ѳ - cot2Ѳ sec2Ѳ - cosec2Ѳ = tan2Ѳ - cot2Ѳ co s B co s A 2 Example: Prove that tan A tan B 2 2 2 2 2 co s A . co s B 2 2 Sol: Given that tan A tan B sec A 1 (sec B 1) 2 2 sec A sec B 2 2 1 2 co s A 1 2 co s B co s B co s A 2 2 2 2 co s A . co s B Values of the trigonometrical ratios : Sin 00 300 450 600 900 0 1 1 3 1 2 Cos Tan 1 0 3 2 1 2 1 2 1 3 Cosec ∞ 2 2 2 1 0 2 3 2 ∞ 1 3 Sec 1 2 3 Cot ∞ 3 2 1 2 ∞ 1 0 3 Example: find the value of tan450.sec300 - cot900.cosec450 Sol: Given that tan450.sec300 -- cot900.cosec450 =1. 2 -- 0. 2 = 3 2 3 Example: If cosѲ = 3/5, find the value of the other ratios Sol: Given that cosѲ = 3/5 = adj / hyp thus using reference triangle adj = 3, hyp = 5,by Pythagorean principle opp = 4 opp 4 hyp 5 sin sec hyp 5 adj 3 5 cos ec tan hyp 5 opp 4 opp 4 adj cot adj opp 3 4 4 Ѳ 3 3 THANK YOU