Question

Sin(pi/4 - α) , если sinα= √2/3 и pi/2<α

Answer 1

$cos \alpha =- \sqrt{1- \frac{2}{9} }=- \frac{ \sqrt{7} }{3} \\sin( \frac{ \pi }{4} - \alpha )=sin\frac{ \pi }{4}cos \alpha -sin \alpha cos\frac{ \pi }{4}= \frac{ \sqrt{2} }{2} (cos \alpha -sin \alpha )=\\= -\frac{ \sqrt{2} }{6} (\sqrt{7} + \sqrt{2} )$

Answer 2

To find the value of sin(pi/4 - α) when sinα = √2/3 and pi/2 < α, we can use trigonometric identities to simplify the expression.

We know that: sin(pi/4) = 1/sqrt(2)

Now, let's use the trigonometric addition formula for sine: sin(A - B) = sinA * cosB - cosA * sinB

In this case, A = pi/4 and B = α: sin(pi/4 - α) = sin(pi/4) * cos(α) - cos(pi/4) * sin(α)

Now, substitute the values: sin(pi/4 - α) = (1/sqrt(2)) * cos(α) - (1/sqrt(2)) * sin(α)

Next, we need to find the value of cos(α). We can use the Pythagorean identity to find it: cos²(α) + sin²(α) = 1

Given sin(α) = √2/3, we can solve for cos(α): cos²(α) + (√2/3)² = 1 cos²(α) + 2/3 = 1 cos²(α) = 1 - 2/3 cos²(α) = 1/3

Taking the square root of both sides (and considering that α is in the second quadrant where cos(α) is negative): cos(α) = -sqrt(1/3) = -1/sqrt(3)

Now, substitute this value back into the expression: sin(pi/4 - α) = (1/sqrt(2)) * (-1/sqrt(3)) - (1/sqrt(2)) * (√2/3)

Finally, simplify: sin(pi/4 - α) = -sqrt(2)/ (sqrt(2) * sqrt(3)) - √2 / (sqrt(2) * sqrt(3)) sin(pi/4 - α) = -√2 / √(2 * 3) sin(pi/4 - α) = -√2 / √6 sin(pi/4 - α) = -√2 * √6 / (√6 * √6) sin(pi/4 - α) = -√12 / 6 sin(pi/4 - α) = -√3 / 3

So, the value of sin(pi/4 - α) is -√3 / 3.