More from VASUDEVA KH

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KHV’s SYNOPSIS: TRIGNOMETRY-FORMULA AND CONCEPTS, , AN ANGLE: An angle is the amount of rotation of a revolving line w.r.t a fixed straight line (a figure, T, sid erm, e( ina, ar l, m, ), , formed by two rays having common initial point.) The two rays or lines are called the sides of the angle and, common initial point is called the vertex of the angle., Rotation of the initial arm to the terminal arm generates the angle., +ve, , •, Rotation can be anti clock wise or clockwise., angle, •, Angle is said to be +ve if rotation is anti clockwise., •, Angle is said to be -ve if rotation is clockwise., Initial, -ve angle, side(arm), , UNITS OF MEASUREMENT OF ANGLES:, 1 degree = 60'=3600'', a) Sexagesimal system:, , 1, degree, 60, 1, 1, =, 1 second=, degree, 60×60 3600, , In sexagesimal system of measurement,, the units of measurement are degrees, minutes and, seconds., 1 right angle =90 degrees(90 o);, 1 degree = 60 minutes (60'), 1 minute = 60 seconds (60''), , 1 minute=, , b) RADIAN OR CIRCULAR MEASURE : In this system units of measurement is radian., , A radian is the measure of an angle subtended at the center of a circle by an arc whose length is equal, to the radius of the circle. one radian is denoted by 1 c, , 1 radian =570 161 22'', , D, , Do You know?, When, no, unit, is, mentioned with an angle,, it is understod to be in, radians. If the radius of, the circle is r and its, circumference is C then, C=2πr, C/2r =π, for any circle, Circumference/diameter, =π which is constant., π =3.1416(approximately), , A radian is a Constant angle. And,  radians = 1800, , B, Arc, , AB be the Arc, Let the length, of the arc =OA=radius, , ----r----- A, , angle AOB =1 radian, RELATIONSHIP BETWEEN DEGREES AND RADIANS:, 180 o,  radians =180o 1 radian= 1c =, , 1c = 570 17' 45''; 10 =, , Radian measure=, , , o, , , 180, , o, , radian=0.01746 radian, (approximately), , x Degree measure i.e. To convert degrees into radians Multiply by, , 180, 180 o, Degree measure=, x Radian measure. i.e. To convert radians into degrees Multiply by, , , , 180 o, 180 o, , , NOTE: 1. Radian is the unit to measure angle 2. It does not means that π stands for 1800 , π is real number,, where as π c stands for 1800, , LENGTH OF ARC OF A CIRCLE:, If an arc of length “s” subtends an angle θ radians at the center of a circle of radius 'r', then, S =r θ i.e. length of arc = radius x angle in radians (subtended by arc), No of radians in an angle subtended by an arc of circle at the centre =, 1c(1 radian) =, , arc length of magnitude of r, radius of r̊, , 1, , arc, S, =, radius, r

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TO DETERMINE THE VALUES OF OTHER TRIGNOMETRIC RATIOS WHEN ONE, TRIGNOMETRIC RATIO IS GIVEN:If one of the t-ratio is given , the values of, other t-ratios can be obtained by constructing a right angled triangle and using the trigonometric identities, given above, For acute angled traingle, we can write other t ratios in terms of given ratio:, , cosθ=, , 1−s, , cosecθ=, , 1, , perp, s, =, hyp, 1, , Let sinθ=s=, , 1−sin , 2, , =, , 2, , s, , sin , , ; tanθ=, ,  1−sin , 2, ,  1−sin , , 1, , ; secθ=, ,  1−sin , 2, , ;, , 1−s, , 2, , 2, , 1, ; cotθ=, sin , , sin , , We can express sinθ in terms of other trigonometric functions by above method:, , tan , , 1, sinθ= 1−cos  =, =, =, 2,  1tan  cosec , 2, ,  sec −1, ,  1tan, , 2, , sec , , =, , 2, , , , tan , , , , , , −2  2, according as θ ∈Q1 or θ ∈Q2, 3, 3 4, We can find other ratios by forming a rightangled traingle. Let tanθ=4/3,  , ,, 2, , cosθ=, , 1−, , 1, or 9, , 1−, , 1, 9, , ie., , 2 2, 3, , 3, , 1, , For ex. sinθ=1/3, since sine is +ve in Q1 and Q2(II quadrant), we have, or, , 5, 3, , 4, −3, ; cosθ=, 5, 5, , then since in Q3, sine and cosine both are negative, we have sinθ=-, , TRIGNOMETRIC RATIOS OF STANDARD and QUANDRANTAL ANGLES:, Radians, , , 6, , 0, Degrees, , 0 300, , sinθ, 0, cosθ, , , 3, , 450, , 600, , 1, 2, , 1, 2, , 3, , 3, , 1, 2, , 1, 2, , 1, , 2, , 0, , 1, 3, , tanθ, , , 4, , 900, , 2, , 1, , 3, , , , , 2, , 1800, 1, 0, , 2, , 3, 2, 2700, 0, , ∞, , 3600, -1, , -1, , , 12, , 0, , 150, 0, 1, , ∞, 0, , 3, , 0, ,  3−1, 2 2,  31, 2 2, 2−3, , 5, 12, 750, ,  31, 2 2,  3−1, 22, 2 3

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VALUES OF T-FUNCTIONS OF SOME FREQUENTY OCCURING ANGLES., Radians 0, , 3, 4, , 2, 3, 1200, , Degrees, , 1350, , 3, , sinθ, , −, , 1, 2, , −, , 0, , −, , -1, ,  ), , (any, , (-1)n, , 3, −, , 1, 2, , n, , , 2, , , 2, , (odd ), , 1, 2, , −3, , tanθ, , 2n1 , , 1500, , 1, 2, , 2, , cosθ, , 5, 6, , (-1)n, , 2, , 0, , 1, 3, , ∞, 0, , RATIOS OF GENERAL ANGLES:, 1. i)cos n  `=(-1)n , ii) sin n  =0, 2. i) sin n, , , = 0 if n is even, 2, , iii) tan nΠ=0 for n-0, 1,2......., , 3. cos n, , , = 0 if n is odd, 2, , = ±1 if n is odd, e.g: sin, sin(, , , 5, = sin, =sin, 2, 2, 3, ) = sin, 2, , cos(odd, , , )=0;, 2, , = ±1 if n is even, 9, =.......=1, 2, , 7, 11 , = sin, = ..........=-1, 2, 2, , cos( odd, , , , )=-1, cos(even, ,  ) =1, , cos 2n−1   =0, cos( 2n-1)  =-1, cos(2n  ) =1, 2, DOMAIN AND RANGE OF TRIGNOMETRIC FUNCTIONS:, Function, , Domain, , Range, , sine, , R, , [-1, 1], , cosine, , R, , [-1, 1], , tangent, cotangent, secant, cosecant, , R-{(2n+1), , , }: nε Z, 2, , R-{n  }; nεZ, R-{(2n+1), , , }: nε Z, 2, , R-{n  }; nεZ, , 4, , R, R, (- ∞ ,-1] υ [1,, (- ∞ ,-1] υ [1,, , ∞ ), ∞ )

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Short cut: Supposing we have to find the value of t- ratio of the angle θ, Step1: Find the sign of the t-ratio of θ , by finding in which quadrant the angle θ lies using, QUADRANT RULE ., Step 2: Find the numerical value of the t-ratio of θ using the following method:, t-ratios of θ=, t- ratio of (1800- θ ) with proper sign if θ lies in the second, quandrant, e.g.: cos1200 = -cos600 = -1/2, t-ratio of ( θ -180) with proper sign if θ lies in the third, quandrant, e.g: sin2100 = -sin300 = -1/2, t-ratio of (360- θ ) with proper sign if θ lies in the fourth, quandrant, e.g: cosec3000= -cosec600 = −, , 2, 3, , t-ratio of θ-n (3600 ) if θ>3600, d), If θ is greater than 3600 i.e. θ =n.3600 +α , then remove the multiples of 3600 (i.e. go on, subtracting from 3600 till you get the angle less than 3600 ) and find the t-ratio of the remaining angle, by applying the above method. e.g: tan10350 =tan6750 (1035-360) =tan3150 = -tan450 =-1, , COMPLIMENTARY AND SUPPLIMENTARY ANGLES:, If θ is any angle then the angle, , , - θ is its complement angle and the angle, 2, ,  - θ is its, , supplement angle., a) trigonometric ratio of any angle = Co-trigonometric ratio of its complement, sin θ = cos(90- θ ), cos θ = sin(90- θ ), tan θ = cot(90- θ ) e.g. sin600 =cos300 , tan600 =cot300 ., b) sin of(any angle) = sin of its supplement ; cos of ( any angle) = -cos of its supplement, tan of any angle = - tan of its supplement i.e. sin 300 =sin 1500 , cos 600 =-cos 1200, , CO-TERMINAL ANGLES: Two angles are said to be co terminal angles , if their terminal sides, , are one and the same. e.g. θ and 360+ θ or θ and n.360+ θ ; - θ and 360- θ or - θ and n.360- θ are co, terminal angles : a) Trig functions of θ and n.360+ θ are same b) Trig functions of -θ and n.360- θ are, same, , TRIGNOMETRIC RATIOS OF NEGETIVE ANGLES: For negative angles always use :, sin(- θ ) = -sin θ cos(- θ ) = cos θ, tan(- θ )= -tan θ , cosec(- θ )= -cosec θ ; se(- θ ) =sec θ ;, cot(- θ) =sec θ(V.IMP), , COMPOUND ANGLE FORMULAE:, 1. Sin (A + B) = sin A cos B + cos A sin B, 2. sin (A – B) = sin A cos B – cos A sin B, 3. Cos (A + B) = cos A cos B – sin A sin B, 4. cos (A – B) = cos A cos B + sin A sin B, , 5. tan (A + B) =, , tan Atan B , 1 – tan A tan B, , 6. tan (A – B) =, , tan A – tan B , 1tan A tan B, , (A#nπ, B#mπ, A-B#kπ), , 6

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TRIGNOMETRC RATIOS OF HALF, ANGLES-t ratios of sub multiple angles, c)tanθ=, , 2, , , a) sinθ =2sin, cos, =, , 2, 2, 1 tan 2, 2, 2 tan, , , 2, 2 , 1 −tan, 2, 2 tan, , DEDUCTIONS:, , , , 1cos , ; cos2, =, 2, 2, 2, , 1+cosθ=2cos2, , , , , -sin2, =2cos2, -1, 2, 2, 2, 1−tan 2 θ, , 2, =1-2sin 2, =, 2, 2, 1+tan θ, 2, , b) cosθ=cos2, , 1-cosθ=2sin2, , , , 1−cos , ; sin2, =, 2, 2, 2, , Transformation formulae:, , a), , SUMS AND DIFFERENCE TO PRODUCT FORMULAE:, , To convert sum of t- functions into product , following formulae is used, Formula that express sum or difference into products, , Sin C + sin D =, Cos C + cos D =, , 2sin, , CD, C–D, cos, 2, 2, , Sin C – sin D =, , 2cos, , CD, C–D, sin, 2, 2, , CD, C–D, cos, 2, 2, , Cos C – cos D =, , 2sin, , CD, D−C, sin, 2, 2, , 2cos, , or, , −2sin, , CD, C−D, sin, 2, 2, , b) PRODUCT-TO-SUM OR DIFFERENCE FORMULAE :, formula which express products as sum or Difference of sines and cosines., To convert product of t-functions into sum or difference , following formulae is, useful., 2 sin A cos B = sin (sum) + sin (diff) i.e 2 sinA cosB = sin(A+B) + sin(A-B), sinA cosB =, , 1, [sin(A+B) + sin(A-B)], 2, , 2 cos A sin B = sin (sum) – sin (diff) i.e 2 cosA sinB = sin(A+B) – sin(A-B), cosA sinB =, , 1, [sin(A+B) – sin(A-B)], 2, , 2 cos A cos B = cos (sum) + cos (diff) i.e. 2 cosA.cosB = cos(A+B)+cos(A-B), cosA.cosB =, , 1, [cos(A+B)+cos(A-B)], 2, , 2 sin A sin B = cos (diff) – cos (sum) i.e. -2 sinA.sin B = cos(A+B)-cos(A-B), OR 2 sinA.sin B = cos(A-B)-cos(A+B), sinA.sin B =, , −, , 1, [cos(A+B)-cos(A-B)], 2, , 8

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EXPRESSION FOR Sin(A/2) and cos(A/2) in terms of sinA:, , , , , sin, , A, A, cos, 2, 2, , sin, , A, A, −cos, 2, 2, , , , , 2, , =1+sinA, , so that, , sin, , 2, , =1-sinA so that, , sin, , A, A, cos, = ±1 sinA, 2, 2, A, A, −cos, = ±1 −sinA, 2, 2, , By addition and subtraction, we have, 2 sin, , A, = ±1 sinA ± ±1 −sinA ; 2, 2, , Using suitable signs , we can find, , cos, sin, , A, 2, , A, = ±1 sinA, 2, , ,, , cos, , ±1 −sinA, , ∓, , A, 2, , VALUES OF TRIGNOMETRICAL RATIOS OF SOME IMPORTANT ANGLES :, , Angle, →, , 1, 2, , 7, , 150, , 180, ,  3−1, 2 2, ,  5−1, ,  31, 2 2, , 0, , 22, , 1, 2, , 360, , 0, , 750, , Ratio, ↓, ,  8−2 6−2 2, , sin, , 4, , or, ,  31, 2 2, , 1, 1,  102  5  22, 4, 2, , 1,  51 , 4, ,  3−1, 2 2, , 3, ,  25−10 5, , 2−1, , 5−2  5, , 3, , 5 2 5 , , 21, , , , , ,  4−2  2, , 5−1, , 4, ,  4− 6− 2, 2 2, ,  82 6−2 2, , cos, , 1,  10−2  5, 4, , 1,  2−2, 2, , 4, , or, ,  4 6 2, 2 2, , tan, , 6− 4−32, , 2-, , or, , 5, ,   3− 2 2−1 , , cot, , 6± 4±32, , or, , 2+, ,   3 2 21 , sec, , 16−10  28  3−6 ( 6 6−2, , ), , 2 −, , 9, , 2, , 5, , 1, , 2, 5, , , , 2+, , 3, , 2-, , 3, , 62

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RELATION BETWEEN THE SIDES & ANGLES OF A TRIANGLE:, The angles of traingle ABC are denoted by A,B, and C. a,b, and c are respectively the sides, opposite to the angles A,B and C. In any traingle ABC , the following results or rule hold, good., , a, b, c, =, =, =2R Where, sinA sinB sinC, , 1 Sine rule’: a = 2R sin A, b = 2R sin B, c = 2R sin C ie, , R is the circum radius of circum circle that passes through the vertices of the traingle., cos A =, ,  b2 c2 – a2 , 2bc, , b2 =a2 +c2 -2ac cosB or, , cos B =, ,  c a – b , 2ca, , c 2 =a2 +b2 -2ab cosC or, , cos C =, ,  a b – c , 2ab, , 2, , 2, , 2, , 2.‘Cosine rule’: a =b +c -2bc cosA or, , 2, , 2, , 2, , 2, , 2, , 2, , 3.Projection rule’:, a = b cos C + c cos B; b = c cos A + a cos C; c = a cos B + b cos A, 6. Formula that involve the Perimeter: If S=, , abc, , where a+b+c is the, 2, , perimeter of a traingle, R the radius of the circumcircle, and r the radius of the inscribed, circle, then, 6. Area of traingle: ∆=, ∆=, ,  s s−a s−b s −c , , ;(HERO'S FORMULA), , 1, 1, 1, abc, a.b.SinC =, b.c. sinA =, c.a.sinB=, 2, 2, 2, 4R, 2, , ∆=, , 1 a sinB. sinC, 1 a 2 sinB. sinC, 1 b 2 sin.C sinA, 1 c 2 sinA. sinB, =, =, =, 2, sinA, 2, sinB, 2, sinC, 2 sinBC , , 11