Question

Cos (arcsin1/3 -arccos2 /3)​

Answer 1

соs(a - b) = cos(a)*cos(b)  + sin(a)*sin(b)

sin(arccos(x)) = √(1 - x²)   |x|<=1

cos(arcsin(x)) = √(1 - x²)   |x|<=1

cos(arcsin(1/3) - arccos (2/3)) = cos(arcsin(1/3))*cos(arccos(2/3)) + sin(arcsin(1/3))*sin(arccos(2/3)) = √(1 - 1/3²)*2/3 + 1/3*√(1 - (2/3)²) = √8/3*2/3 + 1/3*√5/3 = 4√2/9 + √5/9 = (4√2 + √5)/9

Answer 2

Ответ:

Объяснение:

1/3=x; 2/3=y

сos(arcsinx-arccosy)=cos(arcsinx)×cos(arccosy)+sin(arcsinx)×sin(arccosy)

cos(arcsinx)=√(1-x²)=$\sqrt{1-\frac{1}{9} } =\sqrt{\frac{8}{9} } =\frac{2\sqrt{2} }{3}$; cos(arccosy)=y=$\frac{2}{3}$;  sin(arcsinx)=x=$\frac{1}{3}$;   sin(arccosy)=√(1-y²)=$\sqrt{1-\frac{4}{9} } =\sqrt{\frac{5}{9} } =\frac{\sqrt{5} }{3}$

cos (arcsin1/3 -arccos2 /3)=$\frac{2\sqrt{2} }{3} *\frac{2}{3} +\frac{1}{3} *\frac{\sqrt{5} }{3} =\frac{4\sqrt{2}+\sqrt{5} }{9}$

Answer 3

To calculate the value of the expression cos(arcsin(1/3) - arccos(2/3)), we can use the trigonometric identities and simplify step by step.

1. Let's start by evaluating the individual components inside the parentheses.

   a) arcsin(1/3): arcsin(1/3) represents the angle whose sine is 1/3. This means that sin(arcsin(1/3)) = 1/3.

   b) arccos(2/3): arccos(2/3) represents the angle whose cosine is 2/3. This means that cos(arccos(2/3)) = 2/3.

2. Now, we have: cos(arcsin(1/3) - arccos(2/3)) = cos(arcsin(1/3)) * cos(arccos(2/3)) + sin(arcsin(1/3)) * sin(arccos(2/3)) = (1/3) * (2/3) + (1/3) * √(1 - (2/3)^2) = 2/9 + 1/3 * √(1 - 4/9) = 2/9 + 1/3 * √(5/9) = 2/9 + √5/27

Therefore, the value of the expression cos(arcsin(1/3) - arccos(2/3)) is 2/9 + √5/27.