Question

If a=5° then (sin²a+sin² 2 a +sin² 3 a+ to 18 terms) is equal to
Answer the question with steps
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Answer 1

Step-by-step explanation:

Given, in △ ABC, DE∥BC

∴ AD/DB = AE/EC [Using Basic proportionality theorem]

⇒1.5/3 = 1/EC

⇒EC = 3/1.5

EC = 3×10/15 = 2 cm

Hence, EC = 2 cm.

(ii) Given, in △ ABC, DE∥BC

∴ AD/DB = AE/EC [Using Basic proportionality theorem]

⇒ AD/7.2 = 1.8 / 5.4

⇒ AD = 1.8 ×7.2/5.4 = (18/10)×(72/10)×(10/54) = 24/10

⇒ AD = 2.4

Hence, AD = 2.4 cm.

Answer 2

1,3,5,⋯,85,87,89is an arithmetic series with first term a = 1

and common difference d = 2

an=a+(n−1)d⟹89=1+(n−1)2⟹2n−1=89⟹n=45

There are 45 terms in the sequence

sin21∘+sin23∘+sin25∘+⋯+2sin285∘+sin287∘+sin289∘

=(1−cos2∘2)+(1−cos6∘2)+(1−cos10∘2)+⋯

+(1−cos170∘2)+(1−cos174∘2)+(1−cos178∘2)

=45(12)−12(cos2∘+cos6∘+cos19∘+⋯+cos170∘+cos174∘+cos178∘)

=452−12[(cos2∘+cos178∘)+(cos6∘+cos174∘)+(cos10∘+cos170∘)+⋯

+(cos86∘+cos94∘)+cos90∘]

=452−12[(2cos90∘cos88∘)+(2cos90∘cos84∘)+(2cos90∘cos80∘)+⋯

+(2cos90∘cos4∘)+cos90∘]

=452=2212