If cos 9 alpha = sin alpha 9 alpha < 90 degree then 5 alpha =
To find the value of 5 alpha, we need to use the given information that cos(9 alpha) = sin(alpha), where the angle 9 alpha is less than 90 degrees.
We can use the Pythagorean identity to rewrite sin squared alpha:
sin^2 alpha = 1 - cos^2 alpha
Now, let's substitute the given information:
1 - cos^2(9 alpha) = sin^2 alpha
Using the Pythagorean identity for the function cos squared, we have:
1 - cos^2(9 alpha) = 1 - sin^2(9 alpha)
Since both sides of the equation are equal to sin^2 alpha, we can set them equal to each other:
1 - sin^2(9 alpha) = sin^2 alpha
Now, let's simplify:
1 = 2 sin^2 alpha
Dividing both sides by 2:
1/2 = sin^2 alpha
Taking the square root of both sides (note that we are assuming alpha is a positive angle):
√(1/2) = sin alpha
√(1/2) can be written as 1/√2 or √2/2:
1/√2 = sin alpha
This means that sin alpha is equal to 1/√2 or √2/2.
To find the value of alpha, we can use the unit circle or a calculator's inverse sine function (sin^(-1)). The inverse sine of 1/√2 or √2/2 is 45 degrees (π/4 radians).
Now that we know the value of alpha is 45 degrees (π/4 radians), we can find the value of 5 alpha:
5 alpha = 5 * 45 degrees = 225 degrees
Therefore, 5 alpha is equal to 225 degrees.