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Topic : Allied Angles, 07 Month 2020
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Learning Objective/ Key learning, ►, , Page 4, , Apply the concept of allied angles to solve the given simple engineering problem(s)., , Maharashtra State Board of Technical Education, , 25 August 2020
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Contents, ►, ►, ►, , Page 5, , Definition of allied angles, Trigonometric ratios of allied angles, Examples based on allied angles, , Maharashtra State Board of Technical Education, , 25 August 2020
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Allied angles, Definition : If the sum or difference of the measures of two angles is either zero or is an integral multiples of, 𝜋, 90, that is n. then these angles are called allied angles., 2, , For any angle , let be It’s allied angle then, , and, , 𝜋, 2, 𝜋, n., 2, , 𝜋, 2, 𝜋, n., 2, , +=n, , = n. – allied angle of , , –=, , =, , + allied angle of ., , 𝜋, 2, , For any angle of ; n. are allied angles of ., Note: Trigonometric Ratios of Negative angles, sin(- 𝜃 )= -sin 𝜃 and cos(- 𝜃 ) = cos 𝜃, Trigonometric Ratios of Allied angles, If n is an even integer, 𝜋, 2, 𝜋, 𝑛, 2, , , , sin 𝑛 ± 𝜃 = ± sin𝜃, , , , cos, , Page 6, , ± 𝜃 = ± cos𝜃, , Maharashtra State Board of Technical Education, , 25 August 2020
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in If n is an odd integer, 𝜋, 2, 𝜋, 𝑛, 2, , , , sin 𝑛 ± 𝜃 = ± cos𝜃, , , , cos, , ± 𝜃 = ± sin𝜃, 𝜋, 2, , The algebraic sign is settled down by knowing the quadrant in which the angle 𝑛 ± 𝜃 lies, Note that, sin n𝜋 =0, Cos n𝜋 = – 1 If n is odd integer., Cos n𝜋 = 1, If n is Even integer., , 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, Page 7, , Maharashtra State Board of Technical Education, , tan A+ tan B, , 25 August 2020
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Solved Examples:, 1) Without using calculator, find the value of, i) sin( 765) ii) sec (3660), Solution:, i) sin ( 765), sin (– ) = – sin , = sin ( 765), = sin (8 90 + 45), = sin 45, 1, =, 2, , ii) sec (3660), = sec (40 90 + 60), = + sec 60, = 2, , Page 8, , Maharashtra State Board of Technical Education, , 25 August 2020
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2 )Without using calculator, find the value of, sin (420) cos (390) + cos ( 300) sin ( 330), Solution: Given sin 420 cos 390 + cos ( 300) sin ( 330), sin (420) = sin (4 90 + 60), = sin 60, 3, 2, cos (390) = cos (360 + 30), = cos (4 90 + 30), = cos 30, =, , =, , Page 9, , 3, 2, , Maharashtra State Board of Technical Education, , 25 August 2020
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cos ( 300), , = cos (300), = cos (3 90 + 30), = sin 30, 1, =, 2, sin ( 330) = sin (330), = sin (360 30), = sin (4 90 30), = ( sin 30), = sin 30, 1, =, 2, , Page 10, , Maharashtra State Board of Technical Education, , 25 August 2020
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Given sin 420 cos 390 + cos ( 300) sin ( 330), 3 3 1 1, ., +, 2 2, 2 2, 3 1, = +, 4 4, 3+1, =, 4, 4, =, 4, =1, , =, , Page 11, , Maharashtra State Board of Technical Education, , 25 August 2020
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Summary, , So today we learnDefinition of Allied angle, Trigonometric ratios of allied angles, Solved examples based on allied angles., , ►, ►, ►, , Quiz, 1) Find the value of Sin (210°), a) -, , 1, 2, , b), , 7, 2, , c), , 3, 2, , d), , 5, 2, , 2) Find the value of 𝑆𝑒𝑐 2 (-765°), a) 1 b) 4 c) 7 d) 2, , Ans: 1. a) 2. d), Page 12, , Maharashtra State Board of Technical Education, , 25 August 2020
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Thank You, 23, 2, , Page 13, , Maharashtra State Board of Technical Education, , 25 August 2020
