Trigonometry

Question

The value of sin ² $\pi \div$18 + sin ² $\pi \div$9 +sin ² 7$\pi \div$ 18+ sin ² 4$\pi \div$9 =

1. 1
2. 4
3. 2
4. 8

Answer 1

sin² $\frac{π}{18}$ + sin² π/9 + sin² 7π/18 + sin² 4π/9 

Use these 2 equations to re-write the original problem. 

sin² $\frac{4π}{9}$ = cos² ($\frac{π}{2}$ - $\frac{4π}{9}$) = cos² ($\frac{π}{18}$) .................(1)

sin² $\frac{7π}{18}$= cos² ($\frac{π}{2}$ - $\frac{7π}{18}$) = cos² ($\frac{π}{9}$) ...................(2)
 

From (1) and (2)

sin²$\frac{π}{18}$ + sin² $\frac{π}{9}$ + cos² ($\frac{π}{9}$) + cos² ($\frac{π}{18}$) 

sin² $\frac{π}{18}$ + cos² ($\frac{π}{18}$) + sin² $\frac{π}{9}$ + cos² ($\frac{π}{9}$)  

[sin² $\frac{π}{18}$ + cos² ($\frac{π}{18}$)] + [sin² $\frac{π}{9}$ + cos² ($\frac{π}{9}$)]  

1 + 1 = 2 

Hence,Option(3)2 is the correct choice.

Answer 2

Given:

sin^2(pi/18)+sin^2(pi/9)+sin^2(7pi/18)+sin^2(4pi/9)

sin^2(pi/18+pi/9+7pi/18+4pi/9)

sin^2(8pi/18+5pi/9)

sin^2(8pi/18+10pi/18)

sin^2(18pi/18)=sin^2 pi(i.e,1-cos^2 pi)(value of pi=180)

1-cos^2pi=1-(-1)=2 so the option 3 is the correct answer