Question

Arccos√2\2-arcsin√3\2

Answer 1

$arccos \frac{ \sqrt{2} }{2} - arcsin \frac{ \sqrt{3} }{2} = \frac{ \pi }{4} - \frac{\pi}{3} = \frac{3\pi - 4\pi}{12} = \frac{-\pi}{12}$

Answer 2

To simplify the expression arccos(√2/2) - arcsin(√3/2), we can use the known values of trigonometric functions at certain angles.

First, let's look at the value of √2/2. This is the cosine of π/4 radians (45 degrees). So, we have:

arccos(√2/2) = arccos(cos(π/4))

Since arccos and cos are inverse functions, they cancel each other out, leaving us with:

arccos(√2/2) = π/4

Next, let's examine the value of √3/2. This is the sine of π/3 radians (60 degrees). So, we have:

arcsin(√3/2) = arcsin(sin(π/3))

Again, since arcsin and sin are inverse functions, they cancel each other out, giving us:

arcsin(√3/2) = π/3

Now, we can substitute these values back into the original expression:

arccos(√2/2) - arcsin(√3/2) = π/4 - π/3

To subtract these fractions, we need a common denominator. The common denominator here is 12. So, we have:

(3π - 4π) / 12 = -π / 12

Therefore, the simplified expression is -π/12.