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TRIGONOMETRIC IDENTITIES • Reciprocal identities 1 1 sin u = cos u = csc u sec u 1 1 tan u = cot u = cot u tan u 1 1 csc u = sec u = sin u cos u • Pythagorean Identities sin2 u + cos2 u = 1 1 + tan2 u = sec2 u 1 + cot2 u = csc2 u • Quotient Identities sin u cos u tan u = cot u = cos u sin u • Co-Function Identities π π sin( − u) = cos u cos( − u) = sin u 2 2 tan( csc( π π − u) = cot u cot( − u) = tan u 2 2 sec( π − u) = csc u 2 • Sum-to-Product Formulas sin u + sin v = 2 sin u+v 2 u+v 2 u+v 2 u+v 2 cos u−v 2 u−v 2 u−v 2 u−v 2 • Power-Reducing/Half Angle Formulas sin2 u = 1 − cos(2u) 2 1 + cos(2u) cos2 u = 2 1 − cos(2u) tan2 u = 1 + cos(2u)

sin u − sin v = 2 cos

sin

cos u + cos v = 2 cos

cos

cos u − cos v = −2 sin

sin

π − u) = sec u 2

• Product-to-Sum Formulas sin u sin v = cos u cos v = sin u cos v = cos u sin v = 1 [cos(u − v) − cos(u + v)] 2 1 [cos(u − v) + cos(u + v)] 2 1 [sin(u + v) + sin(u − v)] 2 1 [sin(u + v) − sin(u − v)] 2

• Parity Identities (Even & Odd) sin(−u) = − sin u cos(−u) = cos u tan(−u) = − tan u cot(−u) = − cot u csc(−u) = − csc u sec(−u) = sec u • Sum & Diﬀerence Formulas sin(u ± v) = sin u cos v ± cos u sin v cos(u ± v) = cos u cos v sin u sin v tan u ± tan v tan(u ± v) = 1 tan u tan v • Double Angle Formulas sin(2u) = 2 sin u cos u cos(2u) = cos2 u − sin2 u = 2 cos2 u − 1 = 1 − 2 sin2 u 2 tan u tan(2u) = 1 − tan2 u

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...sin2x+x=1-sin2x2sinx+sinx-2sin3x sin2x+x=3sinx-4sin3x References: 1: http://www.math.siu.edu/previews/109/109_Topic8.pdf 2 and 3: http://www.vitutor.com/geometry/trigonometry/identities_problems.html 4.) Use the Pythagorean Identity to find cosx, if sinx= -12 and the terminal side of x lies on quadrant III A: cosx= -32 S: sin2x+cos2x=1 (-12) 2+cos2x=1 14+cos2x=1 cos2x=34 cos2x=34 cosx=32 *note that cosine in 3rd quadrant is negative 5.) Use sum and difference identity to find the exact value of sin75 A: sin75= 6+24 S: sin75=sin45+30 sin75=sin45cos30+cos45sin30 sin75=2232+ 2212 sin75= 6+24 6.) Use a half-angle identity to find the exact value of cos15 A: cos15= 2+32 S: cos15=cos302 cos302= 1+cos302 cos15=1+322 cos15= 2+34 References: 4 : Young, Algebra and Trigonometry, Second Edition, Chapter7: Analytic Trigonometry p668. 5 : Young, Algebra and Trigonometry, Second Edition, Chapter7: Analytic Trigonometry p686. 6 : Young, Algebra and Trigonometry, Second Edition, Chapter7: Analytic Trigonometry p704. 7.) Use quotient identity to find tanx and cotx if sinx=35 and cosx= -45 A: tanx=-34 and cotx=......

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...Browse * Saved Papers ------------------------------------------------- Top of Form Bottom of Form * Home Page » * Other Topics History of Indian Mathematics In: Other Topics History of Indian Mathematics MATHEMATICS IN INDIA The history of maths in india is very great & eventful.Indians gave the system of numerals, zero, geometry & equations to the world. The great Indian mathematician Aryabhata (476-529) wrote the Aryabhatiya ─ a volume of 121 verses. Apart from discussing astronomy, he laid down procedures of arithmetic, geometry, algebra and trigonometry. He calculated the value of Pi at 3.1416 and covered subjects like numerical squares and cube roots. Aryabhata is credited with the emergence of trigonometry through sine functions. Around the beginning of the fifteenth century Madhava (1350-1425) developed his own system of calculus based on his knowledge of trigonometry. He was an untutored mathematician from Kerala, and preceded Newton and Liebnitz by a century. The twentieth-century genius Srinivas Ramanujan (1887-1920) developed a formula for partitioning any natural number, expressing an integer as the sum of squares, cubes, or higher power of a few integers. Origin of Zero and the Decimal System The zero was known to the ancient Indians and most probably the knowledge of it spread from India to other cultures. Brahmagupta (598-668),who had worked on mathematics and astronomy, was the head of the astronomy observatory in......

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...Trigonometry Topic Assessment 1. Solve these equations for [pic] a. [pic] b. [pic] c. [pic] [6] 2. Solve these equations for [pic]. Give your answers as a multiple of [pic]. a. [pic] b. [pic] c. [pic] [6] 3. Solve these equations for [pic] a. [pic] b. [pic] c. [pic] [9] 4. Solve these equations for [pic]. Give your answers as a multiple of [pic]. a. [pic] b. [pic] c. [pic] [9] 5. Solve these equations for [pic] a. [pic] b. [pic] c. [pic] [9] 6. Solve these equations for [pic] a. [pic] [4] b. [pic] [4] c. [pic] [3] Total 50 marks Topic Assessment Solutions 1. (i) [pic] Solutions are in the 1st and 4th quadrants. [pic] or [pic] [pic] (ii) [pic] Solutions are in the 3rd and 4th quadrants [pic] or [pic] [pic] (iii) [pic] Solutions are in 1st and 3rd quadrants. [pic] or [pic] [pic] 2. (i) [pic] Solutions are in 1st and 4th quadrants. [pic] or [pic] [pic] (ii) [pic] Solutions are in 1st and 2nd quadrants [pic] or [pic] [pic] (iii) [pic] Solutions are in 1st and 3rd quadrants [pic] or [pic] [pic] 3. (i) [pic] Solutions are in 1st and 2nd quadrants [pic] or [pic] [pic] ...

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