IIT Level Trigonometry – 100 Questions (Set 4: 61–80)

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61. sinθ + cosθ = a అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = (a²−1)/2
= 1 − (a²−1)²/2
Answer = 1 − (a²−1)²/2

62. sinθcosθ = a అయితే sin⁶θ + cos⁶θ = ?
Answer = 1 − 3a²

63. sinθ + cosθ = √2 అయితే sin⁶θ + cos⁶θ = ?
Step: sinθcosθ = 1/2
= 1 − 3/4
Answer = 1/4

64. sinθcosθ = 1/4 అయితే sin⁶θ + cos⁶θ = ?
= 1 − 3(1/16)
Answer = 13/16

65. tanθ + cotθ = a అయితే sinθcosθ = ?
Answer = 1/a

66. sinθcosθ = a అయితే tanθ + cotθ = ?
Answer = 1/a

67. tanθ + cotθ = 5 అయితే tanθ = ?
x + 1/x = 5
x² − 5x + 1 = 0
Answer = (5 ± √21)/2

68. tanθ + cotθ = 6 అయితే tanθ = ?
x² − 6x + 1 = 0
Answer = 3 ± 2√2

69. sinθ + cosθ = a అయితే sin³θ + cos³θ = ?
Answer = a³ − (3a(a²−1))/2

70. sinθcosθ = a అయితే sin³θcos³θ = ?
Answer = a³

71. sinθ + cosθ = 1 అయితే sin³θ + cos³θ = ?
Step: sinθcosθ = 0
Answer = 1

72. sinθ + cosθ = 0 అయితే sin³θ + cos³θ = ?
Answer = 0

73. sinθ = 8/17 అయితే cosθ = ?
Answer = 15/17

74. cosθ = 7/25 అయితే sinθ = ?
Answer = 24/25

75. tanθ = 12/5 అయితే sinθ = ?
Answer = 12/13

76. tanθ = 5 అయితే cosθ = ?
Answer = 1/√26

77. sinθ + cosθ max?
Answer = √2

78. sinθcosθ max?
Answer = 1/2

79. tanθ + cotθ min?
Answer = 2

80. sinθ + cosθ = 2 అయితే possible?
Answer = Not possible

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IIT Level Trigonometry – 100 Questions (Set 3: 41–60)

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41. sinθ + cosθ = a అయితే sin²θcos²θ = ?
Step: sinθcosθ = (a²−1)/2
Square → (a²−1)²/4
Answer = (a²−1)²/4

42. sinθcosθ = a అయితే sin⁴θ + cos⁴θ = ?
Answer = 1 − 2a²

43. sinθ + cosθ = √2 అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = 1/2
= 1 − 2(1/4)
Answer = 1/2

44. sinθ + cosθ = 1 అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = 0
Answer = 1

45. sinθcosθ = 1/4 అయితే sin⁶θ + cos⁶θ = ?
Step: = 1 − 3(1/16)
Answer = 13/16

46. sinθ + cosθ = a అయితే sin⁶θ + cos⁶θ = ?
Answer = 1 − 3[(a²−1)/2]²

47. tanθ + cotθ = a అయితే sinθcosθ = ?
Answer = 1/a

48. sinθcosθ = a అయితే tanθ + cotθ = ?
Answer = 1/a

49. tanθ + cotθ = 4 అయితే tanθ = ?
Let x = tanθ
x + 1/x = 4
x² − 4x + 1 = 0
Answer = 2 ± √3

50. tanθ + cotθ = 5 అయితే tanθ = ?
x + 1/x = 5
x² − 5x + 1 = 0
Answer = (5 ± √21)/2

51. sinθ + cosθ = a అయితే sin³θ + cos³θ = ?
Answer = a³ − (3a(a²−1))/2

52. sinθcosθ = a అయితే sin³θcos³θ = ?
Answer = a³

53. sinθ + cosθ = √2 అయితే sin³θcos³θ = ?
Step: sinθcosθ = 1/2
= (1/2)³
Answer = 1/8

54. sinθ = 5/13 అయితే cosθ = ?
Answer = 12/13

55. cosθ = 3/5 అయితే sinθ = ?
Answer = 4/5

56. tanθ = 3/4 అయితే sinθ = ?
Answer = 3/5

57. tanθ = 3 అయితే cosθ = ?
Answer = 1/√10

58. tanθ = 1 అయితే sinθ + cosθ = ?
Answer = √2

59. sinθ + cosθ ≥ ?
Answer = −√2

60. sinθcosθ ≤ ?
Answer = 1/2

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IIT Level Trigonometry – 100 Questions (Set 2: 21–40)

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21. sinθ − cosθ = 0 అయితే sinθcosθ = ?
Step: θ = 45° → sinθcosθ = 1/2
Answer = 1/2

22. sinθ + cosθ = 0 అయితే sinθcosθ = ?
Step: θ = 135° → sinθcosθ = −1/2
Answer = −1/2

23. sin²θ − cos²θ = 0 అయితే θ = ?
Step: sin²θ = cos²θ → θ = 45°
Answer = 45°

24. sinθcosθ = 1/3 అయితే (sinθ + cosθ)² = ?
Step: = 1 + 2/3
Answer = 5/3

25. sinθcosθ = a అయితే sin⁴θ + cos⁴θ = ?
Answer = 1 − 2a²

26. sinθ + cosθ = a అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = (a²−1)/2
= 1 − (a²−1)²/2
Answer = 1 − (a²−1)²/2

27. tanθ + cotθ = 3 అయితే sinθcosθ = ?
Step: = 1/3
Answer = 1/3

28. tanθ + cotθ = a అయితే sinθcosθ = ?
Answer = 1/a

29. sinθcosθ = a అయితే tanθ + cotθ = ?
Answer = 1/a

30. tanθ = 2 అయితే sinθcosθ = ?
Step: = 2/(1+4)
Answer = 2/5

31. tanθ = 1 అయితే sin²θcos²θ = ?
Step: (1/2)²
Answer = 1/4

32. sinθ + cosθ = √2 అయితే sin²θcos²θ = ?
Answer = 1/4

33. sinθ + cosθ = 1 అయితే sin²θcos²θ = ?
Answer = 0

34. sinθcosθ = a అయితే sin⁶θ + cos⁶θ = ?
Answer = 1 − 3a²

35. sinθ + cosθ = a అయితే sin⁶θ + cos⁶θ = ?
Answer = 1 − 3[(a²−1)/2]²

36. sinθ = 3/5 అయితే cosθ = ?
Answer = 4/5

37. cosθ = 5/13 అయితే sinθ = ?
Answer = 12/13

38. tanθ = 4/3 అయితే sinθ = ?
Answer = 4/5

39. tanθ = 5 అయితే sinθcosθ = ?
Answer = 5/26

40. tanθ + cotθ min value?
Answer = 2

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IIT Level Trigonometry – 100 Questions (Set 1: 1–20)

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1. sinθ + cosθ = √2 అయితే sinθcosθ = ?
Step 1: Square both sides
( sinθ + cosθ )² = 2
Step 2: 1 + 2sinθcosθ = 2
Step 3: sinθcosθ = 1/2
Answer = 1/2

2. sinθ + cosθ = 1 అయితే sinθcosθ = ?
Step: 1 + 2sinθcosθ = 1
sinθcosθ = 0
Answer = 0

3. sin²θ + cos²θ + 2sinθcosθ = ?
Step: = (sinθ + cosθ)²
Answer = (sinθ + cosθ)²

4. sinθcosθ max value?
Step: occurs at 45°
Answer = 1/2

5. sinθ + cosθ max value?
Answer = √2

6. tanθ + cotθ min value?
Step: ≥ 2
Answer = 2

7. tanθ + cotθ = 2 అయితే θ = ?
Step: tanθ = 1
Answer = 45°

8. sinθcosθ = 1/4 అయితే (sinθ + cosθ)² = ?
Step: = 1 + 2(1/4)
Answer = 3/2

9. sin⁴θ + cos⁴θ = ?
Step: = 1 − 2sin²θcos²θ
Answer = 1 − 2sin²θcos²θ

10. sinθcosθ = a అయితే sin⁴θ + cos⁴θ = ?
Answer = 1 − 2a²

11. sinθ + cosθ = a అయితే sinθcosθ = ?
Answer = (a² − 1)/2

12. sinθcosθ = a అయితే sin⁶θ + cos⁶θ = ?
Answer = 1 − 3a²

13. sinθ + cosθ = √2 అయితే sin³θ + cos³θ = ?
Step: = √2/2
Answer = √2/2

14. sinθcosθ = 1/2 అయితే sin³θcos³θ = ?
Answer = 1/8

15. tanθ = 3 అయితే sinθcosθ = ?
Step: = tanθ / (1 + tan²θ)
= 3/10
Answer = 3/10

16. tanθ = a అయితే sinθcosθ = ?
Answer = a/(1+a²)

17. sin²θcos²θ max value?
Answer = 1/4

18. sinθ + cosθ min value?
Answer = −√2

19. sinθcosθ ≤ ?
Answer = 1/2

20. sinθ + cosθ = 2 అయితే possible?
Answer = Not possible

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ట్రిగోనమెట్రీ – Step-wise Solved Problems (181–200) | Ultimate Final Level

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181. sinθ + cosθ = a అయితే sinθcosθ = ?
Step: (a² − 1)/2
Final Answer = (a² − 1)/2

182. sinθcosθ = a అయితే sin²θ + cos²θ = ?
Final Answer = 1

183. sinθ + cosθ = √2 అయితే sinθcosθ = ?
Final Answer = 1/2

184. sinθcosθ = 1/2 అయితే sinθ + cosθ = ?
Final Answer = √2

185. sinθ + cosθ = 1 అయితే sinθcosθ = ?
Final Answer = 0

186. sinθ + cosθ = 0 అయితే sinθcosθ = ?
Final Answer = −1/2

187. tanθ + cotθ = a అయితే sinθcosθ = ?
Final Answer = 1/a

188. tanθ + cotθ = 2 అయితే θ = ?
Final Answer = 45°

189. sinθcosθ = a అయితే sin⁴θ + cos⁴θ = ?
Final Answer = 1 − 2a²

190. sinθcosθ = a అయితే sin⁶θ + cos⁶θ = ?
Final Answer = 1 − 3a²

191. sinθ + cosθ = a అయితే sin³θ + cos³θ = ?
Final Answer = a³ − (3a(a²−1))/2

192. sinθcosθ = a అయితే sin³θcos³θ = ?
Final Answer = a³

193. sinθ + cosθ max value?
Final Answer = √2

194. sinθcosθ max value?
Final Answer = 1/2

195. tanθ + cotθ min value?
Final Answer = 2

196. sin²θcos²θ max value?
Final Answer = 1/4

197. sinθ + cosθ ≥ ?
Final Answer = −√2

198. sinθcosθ ≤ ?
Final Answer = 1/2

199. sinθ + cosθ = 2 అయితే possible?
Final Answer = Not possible

200. sinθcosθ = 1 అయితే possible?
Final Answer = Not possible

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Shaktimatha Learning | Ultimate Completion Series

 

ట్రిగోనమెట్రీ – Step-wise Solved Problems (161–180) | Final Boss Level

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161. sinθ + cosθ = a అయితే sin²θcos²θ = ?
Step: sinθcosθ = (a²−1)/2
Square → (a²−1)² / 4
Final Answer = (a²−1)² / 4

162. sinθcosθ = a అయితే sin⁸θ + cos⁸θ = ?
Step: use identity expansion
= 1 − 4a² + 2a⁴
Final Answer = 1 − 4a² + 2a⁴

163. sinθ + cosθ = √2 అయితే sin⁸θ + cos⁸θ = ?
Step: sinθcosθ = 1/2
= 1 − 4(1/4) + 2(1/16)
= 1 − 1 + 1/8
Final Answer = 1/8

164. sinθcosθ = 1/4 అయితే sin⁸θ + cos⁸θ = ?
= 1 − 4(1/16) + 2(1/256)
= 1 − 1/4 + 1/128
Final Answer = 97/128

165. sinθ + cosθ = a అయితే sin⁸θ + cos⁸θ = ?
Step: substitute sinθcosθ = (a²−1)/2
Final Answer = 1 − 4[(a²−1)/2]² + 2[(a²−1)/2]^4

166. sinθcosθ = a అయితే sin⁴θ + cos⁴θ = ?
Final Answer = 1 − 2a²

167. sinθ + cosθ = a అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = (a²−1)/2
= 1 − 2[(a²−1)/2]²
Final Answer = 1 − (a²−1)²/2

168. tanθ + cotθ = a అయితే sinθcosθ = ?
Step: tanθ + cotθ = 1/(sinθcosθ)
Final Answer = 1/a

169. tanθ + cotθ = 3 అయితే sinθcosθ = ?
Step: = 1/3
Final Answer = 1/3

170. tanθ + cotθ = a అయితే tanθcotθ = ?
Step: tanθcotθ = 1
Final Answer = 1

171. sinθ + cosθ = a అయితే tanθ = ?
Step: tanθ = (sinθ)/(cosθ)
Use ratio method
Final Answer = √[(1−cos²θ)/cos²θ]

172. sinθcosθ = a అయితే tanθ + cotθ = ?
Step: = 1/a
Final Answer = 1/a

173. sinθ + cosθ = 1 అయితే sin⁸θ + cos⁸θ = ?
Step: sinθcosθ = 0
Final Answer = 1

174. sinθcosθ = 0 అయితే sin⁸θ + cos⁸θ = ?
Final Answer = 1

175. tanθ + cotθ ≥ ?
Final Answer = 2

176. sinθ + cosθ ≥ ?
Final Answer = −√2

177. sinθcosθ ≤ ?
Final Answer = 1/2

178. sin²θcos²θ ≤ ?
Final Answer = 1/4

179. sinθ + cosθ = a అయితే maximum a?
Final Answer = √2

180. sinθcosθ = a అయితే maximum a?
Final Answer = 1/2

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Shaktimatha Learning | Final Boss Level Series

 

ట్రిగోనమెట్రీ – Step-wise Solved Problems (141–160) | Extreme Level

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141. sinθ + cosθ = a అయితే sinθcosθ = ?
Step: a² = 1 + 2sinθcosθ
sinθcosθ = (a² − 1)/2
Final Answer = (a² − 1)/2

142. sinθcosθ = a అయితే sin⁶θ + cos⁶θ = ?
Step: = (sin²θ + cos²θ)³ − 3sin²θcos²θ(sin²θ + cos²θ)
= 1 − 3a²
Final Answer = 1 − 3a²

143. sinθ + cosθ = √2 అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = 1/2
= 1 − 2(1/4)
Final Answer = 1/2

144. sinθ + cosθ = 1 అయితే sin⁴θ + cos⁴θ = ?
Step: sinθcosθ = 0
= 1
Final Answer = 1

145. sinθcosθ = 1/4 అయితే sin⁶θ + cos⁶θ = ?
Step: = 1 − 3(1/16)
= 13/16
Final Answer = 13/16

146. sinθ + cosθ = a అయితే sin⁶θ + cos⁶θ = ?
Step: sinθcosθ = (a²−1)/2
= 1 − 3[(a²−1)/2]²
Final Answer = 1 − 3(a²−1)²/4

147. sin²θ + cos²θ + sinθcosθ = ?
Step: = 1 + sinθcosθ
Final Answer = 1 + sinθcosθ

148. sinθ + cosθ = √3 అయితే possible?
Step: max √2 < √3
Final Answer = Not possible

149. sinθcosθ = 3/4 అయితే possible?
Step: max value 1/2
Final Answer = Not possible

150. tanθ + cotθ ≥ ?
Step: ≥ 2
Final Answer = 2

151. tanθ + cotθ = 2 అయితే θ = ?
Step: tanθ = 1
Final Answer = 45°

152. tanθ + cotθ = 4 అయితే tanθ = ?
Let x = tanθ → x + 1/x = 4
Solve → x = 2 ± √3
Final Answer = 2 ± √3

153. sinθ + cosθ = a అయితే sin³θ + cos³θ = ?
= a³ − 3a[(a²−1)/2]
Final Answer = a³ − (3a(a²−1))/2

154. sinθcosθ = a అయితే sin³θcos³θ = ?
Step: = a³
Final Answer = a³

155. sinθ + cosθ = √2 అయితే sin³θcos³θ = ?
Step: sinθcosθ = 1/2
= (1/2)³
Final Answer = 1/8

156. sin²θcos²θ = 1/16 అయితే sinθcosθ = ?
Final Answer = 1/4

157. sinθ + cosθ = 0 అయితే sin³θ + cos³θ = ?
= 0 − 3×(−1/2)×0 = 0
Final Answer = 0

158. sinθ − cosθ = 1 అయితే sinθcosθ = ?
Step: square → 1 = 1 − 2sinθcosθ
sinθcosθ = 0
Final Answer = 0

159. sinθ + cosθ = 1 అయితే θ values?
Final Answer = 0° or 90°

160. sinθcosθ = 0 అయితే θ values?
Final Answer = 0° or 90°

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Shaktimatha Learning | Extreme Level Practice