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