Let ABCDE is a regular pentagon with all sides equal to 4. Which one of the following is a correct solution for the length AC? (i) 2 csc(18°) (ii) 2 sec (72°) (iii) \sqrt{32 - 32 c o s \left(\right. 108 ° \left.\right)}

Engineering

Mathematics

compound angle Formulae

Question

Let ABCDE is a regular pentagon with all sides equal to 4. Which one of the following is a correct solution for the length AC?
(i) 2 csc(18°)
(ii) 2 sec (72°)
(iii) $\sqrt{32 - 32 c o s (108 °)}$

only (iii) and (i) are correct

all are correct

only (i) and (ii) are correct

only (ii) and (iii) are correct

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using cosine rule
x² = 16 + 16 – 2 · 4 · 4 cos (108°)
= 32(1 – cos 108°)   ⇒ x = 8 sin 54°
∴   x = 8 cos 36° = 2( $\sqrt{5} + 1$ ). Now verify I and II also.
⇒   all correct

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