web analytics

Ex 8.2, 1 - Evaluate: (i) sin 60 cos 30 + sin 30 cos 60 - Trignometric ratios of Specific Angles - Evaluating

  1. Chapter 8 Class 10 Introduction to Trignometry
  2. Serial order wise
Ask Download

Transcript

Ex 8.2, 1 Evaluate the following : (i) sin 60° cos 30° + sin 30° cos 60° We know that, sin 60° = √3/2 cos 30° = √3/2 sin 30° = 1/2 cos 60° = 1/2 Putting all values sin 60° cos 30° + sin 30° cos 60° = (√3/2)×(√3/2)+(1/2)×(1/2) = (√3 × √3)/(2 × 2)+1/(2 × 2) = 3/4 + 1/4 = (3 + 1)/4 = 4/4 = 1 Hence, sin 60° cos 30° + sin 30° cos 60° = 1 Ex 8.2, 1 Evaluate the following : (ii) 2 tan2 45° + cos2 30° – sin2 60° We know that, tan 45° = 1 cos 30° = √3/2 sin 60° = √3/2 2tan 2 45° + cos2 30° - sin2 60° Putting values = 2 ×(1)2+(√3/2)^2−(√3/2)^2 = 2 + 0 = 2 Ex 8.2, 1 Evaluate the following : (iii) "cos 45°" /"sec 30° + cosec 30°" We know that, "cos 45°" = 1/√2 "sec 30°" = 1/(cos⁡〖30°〗 ) = 1/(√3/2) = 2/√3 Hence, cos⁡〖45°〗/(sec⁡〖30°〗+ 𝑐𝑜𝑠𝑒𝑐 30°) = (1/√2)/((2/√3 + 2/1) ) = 1/√2 × 1/((2/√3 + 2/1) ) = 1/√2×1/((2 × 1 + 2 × √3)/(√3 × 1)) = 1/√2×(√3 × 1)/((2 × 1 + 2 × √3) ) = 1/√2×√3/((2 + 2√3) ) = (1 ×√3)/(√2 × 2(1 + √3) ) = √3/(2√2 (1 + √3) ) = √3/(2√2 (√3 + 1) ) Rationalizing = (√3 (√3 −1))/(2 √2 (√3 + 1) (√3 −1) ) = (3 − √3)/(2√2 ((√3 )2 −12 ) ) = (3 − √3)/(2√2 (3 −1)) = (3 − √3)/(2√2 × 2) Multiplying √2 on numerator and denominator) = ((3 − √3 ))/(4√2 )× √𝟐/√𝟐 = (3√2 − √3 ×√2 )/(4 × 2) = (3√2 −√6 )/8 Ex 8.2, 1 Evaluate the following : "sin 30° + tan 45° – cosec 60°" /"sec 30° + cos 60° + cot 45°" We know that, sin "30°" = 1/2 tan "45°" = 1 cosec 6"0°" = 1/sin⁡〖60°〗 = 1/(√3/2) = 2/√3 Now, (sin⁡〖30°〗 "+" tan 30°" " −〖 cos𝑒𝑐〗⁡〖30°〗)/(sec⁡〖30°〗 "+ " cos 60°" " + co𝑡⁡〖45°〗 ) Putting values = (1/2 + 1 − 2/√3)/(2/√3 + 1/2 + 1) = ((1 × √3 + 1 × 2 × √3 − 2 × 2)/(2 × √3) )/((2 × 2 + 1 × √3 + 1 × 2 × √3 )/(2 × √3)) = (1 × √3 + 1 × 2 × √3 −2 × 2 )/(2 × 2 + 1 × √3 + 1 × 2 × √3) = ((√3 + 2 √3 −4))/(4 +√3 + 2 √3) = (3 √3 − 4)/((3 √3 + 4) ) Rationalisizing = ((3 √3 − 4))/((3 √3 + 4) )×((3√3 − 4))/((3 √3 − 4) ) = ((3 √3 )2 + (4)2 − 2 × 4 ×3 √3)/((3 √3)2 − (4)2) = (32 × (√3 )2 + 16 − 24 √3)/(32 × (√3 )2 − 16) = (9 × 3 + 16 −24 √3)/(9 × 3 − 16) = (27 + 16 − 24 √3)/11 = (43 − 24 √3)/11 Hence, (sin⁡〖30°〗 "+" tan 30°" " −〖 cos𝑒𝑐〗⁡〖30°〗)/(sec⁡〖30°〗 "+ " cos 60°" " + co𝑡⁡〖45°〗 ) = (43 − 24 √3)/11 Ex 8.2, 1 Evaluate the following : "5 cos2 60° + 4 sec2 30° – tan2 45°" /"sin2 30° + cos2 30°" We know that cos "60° = " 1/2 sec "30° = " 1/cos⁡"30°" sec "30° = " 1/(√3/2) sec "30° = " 2/√3 tan "45°" = 1 (5 𝑐𝑜𝑠2 60°+ 4 𝑠𝑒𝑐2 30°−tan2⁡〖45°〗)/(𝑠𝑖𝑛2 30° + 𝑐𝑜𝑠2 30°) Putting values = (5(1/2)^2 + 4(2/√3)^2 − (1)2)/((1/2)^2 + (√3/2)^2 ) = (5/4 + 4 × ( 2 × 2)/(√3 × √3) − 1)/(1/4 + (√3 × √3)/(2 × 2)) = (5/4 + 16/3 − 1)/(1/4 + 3/4) = ((5 × 3 + 16 × 4 − 1 × 4 × 3 )/(4 × 3) )/((1 + 3)/4 ) = ((15 + 64 − 12 )/(4 × 3) )/(4/4 ) = ((67 )/12 )/1 = 67/12 Hence, (5 𝑐𝑜𝑠2 60°+ 4 𝑠𝑒𝑐2 30°−tan2⁡〖45°〗)/(𝑠𝑖𝑛2 30° + 𝑐𝑜𝑠2 30°) = 67/12

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail