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1 VIII_OLYMPIAD_PHYSICS_VOL-1 MATHEMATICS FOR PHYSICS TRIGONOMETRY SYNOPSIS Introduction: The word ‘trigonometry’ is derived from the Greek roots --- ‘tri’ meaning ‘three’; ‘gonia’ meaning ‘an angle’ ; ‘metron’ meaning ‘measure’. Thus ‘trigonometry’ means three angle measure. It is an analytical study of a three angled geometric figure, namely the triangle. Trigonometrical Ratios:In right triangle OPM, if  POM =  , then the side MP (perpendicular) is called the opposite side ; the longest side OP is called the hypotenuse and the third side OM is called the Adjacent side (base) of the triangle. P X O M X I Y YI  P X O M X I Y YI  (i) Sine of angle  , written as sin  = MP OP = Opposite side Hypotenuse side (ii) Cosine of angle  , written as cos  = OM OP = Adjacent side Hypotenuse side (iii) Tangent of angle  , written as tan  = MP OM = Opposite side Adjacent side (iv) Cotagent of angle  , written as cot  = OM MP = Adjacent side Opposite side (v) Secant of angle  , written as sec  = OP OM = Hypotenuse side Adjacent side (vi) Cosecant of angle  , written as cosec  = OP MP = Hypotenuse side Opposite side [Note :i) From the above relations, we can say that sin = 1/cosec  or cos  = 1/sec  ii) According to pythagoras theorem, (Hypotenuse side)2 =( Adjacent side)2 + (Opposite side)2
2 VIII_OLYMPIAD_PHYSICS_VOL-1 Table for the values of Trigonometric ratios of some standard angles : 0 4 1 2 1 2 1 2 1 2 1 3 1 3 3 2 3 2 1 4 2 4 3 4 4 4 3 3 2 2 2 3 2 3 Trigonometric Identities : i) Sin2  + Cos2  = 1 ii) Sec2  – Tan2  = 1 iii) Cosec2  – Cot2  = 1 Signs of T-ratios and Limits to the value of T-ratios In order to understand the signs of T-ratios, let us first understand the rule for signs of the sides of the triangle OMP. (i) OM is positive if it is drawn to the right of O and negative if drawn to the left. p X O M X I Y YI  (ii) MP is positive if it is drawn above XIOX and negative if drawn below XIOX. (iii) OP is always positive.
3 VIII_OLYMPIAD_PHYSICS_VOL-1 Keeping in mind this sign convention and the definitions of T-ratios, we shall find that : (a) In the first quadrant, all the T-ratios are positive. (b) In the second quadrant, only the sine and (its reciprocal) cosecant are +ve. (c) In the third quadrant, only the tangent and (its reciprocal) cotangent are +ve. (d) In the fourth quadrant, only the cosine and (its reciprocal) secant are +ve To find the values of Trigonometric ratios when 0 θ > 90 : [Note: The sign convension of trigonometric functions in the four quadrants are shown in the above figure by using the phrase all silver tea cups]. Every angle A can be reduced to the form: 0 A n.90    where nZ and 0 0 90    . (i) If n is a odd integer, we have       0 0 0 sin n.90 cos , cos n.90 sin , tan n.90 cot                ,       0 0 0 cot n.90 tan, cosec n.90 sec , sec n.90 cosec ,                i.e., sin changes to cos, cos changes to sin, tan changes to cot, cot changes to tan, sec changes to cosec and cosec changes to sec. (ii) If n is an even integer, we have       0 0 0 sin n.90 sin , cos n.90 cos , tan n.90 tan                ,       0 0 0 cosec n.90 cosec , sec n.90 sec , cot n.90 cot                i.e., sin remains sin, cos remains cos, tan remains tan, cosec remains cosec, sec remains sec and cot remains cot. (iii) Use original (given) ratio to find + or – sign in the R.H.S. of the equations in (i) and (ii) making use of the phrase All Silver Tea Cups. Example 1: sin 120° = sin (1.90°+30°). Here n = 1, an odd integer. Sin changes to Cos  120° lies in the second quadrant.  Original (given) ratio sin is + ve.  sin 120° = sin (90°+30°) = +cos 30° = 3/2 . Example 2: cos 240° = cos (180° + 60°) = cos (2.90° + 60°) Here n = 2, an even integer  cos remains cos. 240° lies in the third quadrant  original (given) ratio cos is –ve.  cos 240° = – cos 60° = –1/2. Example 3: tan (–300°) = – tan 300° = –tan (3.90° + 30°). Here n = 3, an odd integer. tan 300° lies in the fourth quadrant  original (given) ratio tan is –ve  tan (–300°) = –(–cot30°) = 3 .
4 VIII_OLYMPIAD_PHYSICS_VOL-1 Some more important formula : 1) Sin 2  = 2 Sin Cos  2) Sin  = 2 Sin(  /2) Cos(  /2) 3) Cos2  = 2 Cos2  – 1 4) Cos  = 2 Cos2 (  /2) – 1 5) Sin (A + B) = Sin A Cos B + Cos A Sin B 6) Sin (A – B) = Sin A Cos B – Cos A Sin B 7) Cos (A + B) = Cos A Cos B – Sin A Sin B 8) Cos (A – B) = Cos A Cos B + Sin A Sin B WORKSHEET CUQ:1.  m m m From figure, the value of cos is [ ] 1) 4/5 2) 3/5 3) 3/4 4) 4/3 2. In the above question, the value of sec is [ ] 1) 3/4 2) 4/3 3) 5/3 4) 4/5 3. In the above question (1), the value of sin is [ ] 1) 4/5 2) 3/5 3) 3/4 4) 4/3 4. In the above question (1), the value of Tan is [ ] 1) 3/4 2) 4/3 3) 5/3 4) 4/5 5. In the above question (1), the value of cot is [ ] 1) 3/4 2) 4/3 3) 5/3 4) 4/5 6. If sin =1/2, then cos ec =_____ [ ] 1) 1/2 2) 2 3) 3 4) 2 7. The product of Sin 300 and Cos 600 is [ ] 1) 1 2 2) 1 4 3) 1 4) 3 4 8. The ratio of Tan 600 and Cot 600 is [ ] 1) 1 2 2) 3 3) 3 4) 1 3 9. Tan 600 – Cot 600 = ________________ [ ] 1) 2 3 2) 1 5 3 3) 1 3 4) 2 3 10. Sin 300 × Cos 300 + Sin 450 × Cos 450 ________________ [ ] 1) 2 3  4 2) 2 3  4 3) 1 3 4  4) 1 3  4