Question

Calculate sin2a if sina+cosa=1/3

Accepted Answer

We have sin a + cos a = 1/3
Use the fact that sin 2a = 2*(sin a)(cos a)
sin a + cos a = 1/3
square the right and left hand sides
(sin a + cos a)^2 = (1/3)^2
=> (sin a)^2 + (cos a)^2 + 2*(sin a)(cos a) = 1/9
(sin a)^2 + (cos a)^2 = 1
=> 1 + 2*(sin a)(cos a) = 1/9
=> 2*(sin a)(cos a) = 1/9 - 1
=> 2*(sin a)(cos a) = -8/9
=> sin 2a = -8/9
The required value of sin 2a = -8/9

Answer

We'll raise the square the given constraint:
(sina+cosa)^2=(1/3)^2
We'll expand the binomial:
(sina)^2 + 2sina*cosa + (cosa)^2 =1/9
But (sina)^2+ (cosa)^2= 1 ( Pythagorean identity )
2sina*cosa+1=1/9
2sina*cosa = 1/9 - 1
We'll apply double angle identity:
sin2a=2sina*cosa
sin2a=1/9 - 1
sin2a = -8/9
The value of sin2a is sin2a = -8/9.

trigonometry math

Calculate sin2a if sina+cosa=1/3

Expert Answers

We’ll help your grades soar