​ If sin θ + cos θ = 1/2, then ​​ 16(sin(2θ) + cos(4θ) + sin(6θ)) is equal to: (1) 23 (2) –27 (3) –23 (4) 27 ​

Question

If sin\(\theta\) + cos \(\theta\) = \(\frac{1}{2}\), then

16(sin(2\(\theta\)) + cos(4\(\theta\)) + sin(6\(\theta\))) is equal to: 

(1) 23 

(2) –27 

(3) –23 

(4) 27

Answer 1

Correct option (3) -23

sin θ + cos θ = \(\frac{1}{2}\)

sin² θ + cos² θ + 2sinθcosθ = \(\frac{1}{4}\)

sin 2θ = \(\frac{-3}{4}\)

Now :

cos 4θ = 1-2sin²θ